Establishing What is Being Stated
The first thing to be established is the meaning of Act 10:43. Note that additional verse references, unless otherwise noted, are also from Acts chapter 10.
I believe to do this, one must at least consider as well v.42. The Greek text with my interlinear translation (largely following the NKJV) is:
42 Καὶ παρήγγειλεν ἡμῖν κηρύξαι τῷ λαῷ, καὶ διαμαρτύρασθαι
and commanded us to preach to the people and to testify
ὅτι αὐτός ἐστιν ὁ ὡρισμένος ὑπὸ τοῦ θεοῦ
that he is the one who has been ordained by the God
κριτὴς ζώντων καὶ νεκρῶν.
Judge of [the] living and dead
43 Τούτῳ πάντες οἱ προφῆται μαρτυροῦσιν,
To Him all the prophets testify to
ἄφεσιν ἁμαρτιῶν λαβεῖν διὰ τοῦ ὀνόματος αὐτοῦ
forgiveness of sins to be received through the name of Him
πάντα τὸν πιστεύοντα εἰς αὐτόν.
all the ones believing in him
 The minority of texts have οὗτός instead of αὐτός here, but that does not change the meaning, for both οὗτός ("this one") or αὐτός ("he") refers back to the subject under discussion, Jesus, from v.36 and following, the ordained Judge noted in v.42.
 Τούτῳ is dative singular, but can be either masculine or neuter gender. If masculine, it refers back to "this one" (i.e. Jesus), if neuter, it refers back to the previous sentence, "this assertion [of an appointed Judge]" being the idea. In context, it probably does not matter much, as even the masculine reference is directly relating back to the "Judge" just noted, which is already referring back to Jesus. What is critical to the discussion here is this judgeship reference.
Besides the notations, it is important here to simply notice the particulars of the phrase πάντες οἱ προφῆται ("all the prophets"), where both πάντες and οἱ προφῆται are plural.
The Subject of Testimony
The subject of the testimony is "to Him" or "to this" (see notes above), which relates to Jesus being made Judge. That is, we are going to be looking for Hebrew Scripture references to a supreme Judge specifically (and best if ordained by God), or judgement by the Messiah (Jesus is considered the Christ, v.36) or the Lord above others (Jesus is considered Lord of all, v.36). Though this last reference most Christians, and arguably Peter, would also be a reference to Jesus being God Himself. Through this Judge's name, forgiveness of sins is received by the act of believing Him (v.43).
Note that to Peter, the "name of him" is the same name as that of God and the Holy Spirit, as preached to Peter by Christ (Mt 28:19), but also note that "Jesus Christ" means "Savior Messiah," so a reference to "Savior" or "Joshua" or "Messiah" would suffice for Hebrew Scripture references if one wanted to get particular. However, "receive through the name of" is a statement about receiving forgiveness on behalf of the Person Himself that one is believing upon, not necessarily the specific "name" of that Person (though the name identifies the proper Person).
The Meaning of "All the Prophets"
There are at least eight possible meanings to "all the prophets." It is critical to note, however, that what is being referred to is the Hebrew writings related to the prophets. This means first of all identifying that body of texts. The Nevi'im ("Prophets") are traditionally split into two categories, each containing four books:
- Former Prophets
- Latter Prophets
- The Twelve (i.e. all the "Minor Prophets" in most English Bibles)
Note: Daniel is considered in the Ketuvim ("Writings") in Hebrew Scriptures, though most Christians consider him more of a prophet as well.
Now, given above organization of Hebrew Scriptures for the Nevi'im, the eight possible meanings of "all the prophets" are:
- A reference to the whole ("all") of the prophetic writings, i.e. the Nevi'im as a whole or even all of Scripture as a whole. This is essentially what one answer's position is.
- A reference to the two divisions, i.e. the Former and Latter Prophets.
- A reference to each of the four books in the Latter Prophets alone.
- A reference to the division of the Former Prophets as a group and the Latter Prophets as four individual books (i.e. 5 sets of writings; see also ).
- A reference to Samuel and the four books in the Latter Prophets (5-6 books total).
- A reference to the division of the Latter Prophets as a group and the Former Prophets as four individual books (i.e. 5 sets of writings; see also ).
- A reference to each book of the Hebrew Scriptures from the Nevi'im, i.e. the eight books noted above.
- A reference to each individual writing of the actual prophets themselves, i.e. not just the books noted, but also including each of the twelve men contained in The Twelve.
 According to William Arndt, Frederick W. Danker, and Walter Bauer, A Greek-English Lexicon of the New Testament and Other Early Christian Literature (Chicago: University of Chicago Press, 2000), s.v. πᾶς, the construction of a plural form of πᾶς with a plural substantive such as οἱ προφῆται falls under the first definition that states "pert. to totality with focus on its individual components, each, every, any" (1.b.β.א). The fourth definition has the idea of "whole," and it is only found with the plural when the πᾶς is found between the plural article and the plural substantive (4.c.β), which is not the formation given in v.43. Note also that part of the argument given in the other answer advocating that view revolves around Luke's usage of the term. While I would agree Luke wrote Acts, I would also maintain that Luke is essentially quoting or at least paraphrasing Peter here (v.34), so it is more what does Peter mean by this phrase than what Luke may mean as a narrator. Additionally, the proof text given of Luke 24:27 seems to argue against it referring to the totality of Scripture, as there is essentially an order given. To requote (NKJV):
And beginning at Moses and all the Prophets, He expounded to them in
all the Scriptures the things concerning Himself.
The Greek majority text is (there are some textual variants here—spelling of Moses' name, aorist [NA28] vs. imperfect for διηρμήνευεν—but not significant to the discussion):
Καὶ ἀρξάμενος ἀπὸ Μωσέως καὶ ἀπὸ πάντων τῶν προφητῶν, διηρμήνευεν
αὐτοῖς ἐν πάσαις ταῖς γραφαῖς τὰ περὶ ἑαυτοῦ.
The participle ἀρξάμενος ("beginning") has two associated start points with joined ἀπὸ phrases: Moses and all the prophets. Now if "all the Scriptures" means the total body of all Scriptures (Torah, Nevi'im, Ketuvim), the implication of "beginning" at two points is showing the order of His exposition: The Law, the Prophets, then the Writings (i.e. the "beginning" refers then to the first two of the three to indicate the order). However, "all the Scriptures" in this context could be a reference to all the writings from those two categories, that is, He began at Moses and expounded from the Torah, then He began at the Prophets and expounded from them (i.e. two beginning points), not covering the Ketuvim at all. In either case, however, the statement of "all the prophets" would not be referring to all of Scripture, and that seems an unlikely interpretation given the construction of the sentence.
 At present, I am unable to establish for certain if this division is historically ancient enough to be considered Peter's meaning.
 At present, I know the Twelve were collated early enough to be considered a single work by Peter's time, for
the first extra-biblical evidence we have for the Twelve as a
collection is c. 190 BCE in the writings of Jesus ben Sirach.
What I have not yet been able to establish is how early the Former Prophets were (1) grouped as an organizational unit (so related to  above) and (2) when they were first referred to themselves as "prophets," nor (3) how early the Latter Prophets were grouped as an organizational unit (also related to ).
 It is possible that in Peter's time, the only real "former" prophet considered so was Samuel (and maybe David, though his role as such overshadowed by his kingship). This is inferring from Act 3:24 especially (cf. 13:20), but also Samuel's (and notably David's) particular separation in Heb 11:32 from "the prophets" otherwise. If so, this would isolate the Books of Samuel (and perhaps Judges, if authored by him) as an earlier understood division of the Former Prophets, being composed by him and possibly David for the later periods.
Given what I state in note  above, I see the #1 choice above as definitively the least likely/defensible choice on a grammatical basis, since it refers to the whole, not any of the component parts. The other seven could be meant, as it depends upon if Peter is referring to in any way or combination of the two component divisions (Former/Latter), Samuel as the only early one, and/or various of the eight components books, or if in fact every one of the prophets within their particular writing. Examination of the Hebrew Scriptures should help isolate Peter's meaning further.
Examining the Prophets
Recall that we are looking for two key components:
- A statement of God's ordaining a Judge (whether explicitly stated or demonstrated that He is given judgment).
- That Judge granting forgiveness to believers (i.e. through His Person whom they are believing upon).
Scripture quotations are from the NKJV:
In chapter 7 we see the Person bearing the name of the Lord (v.9) is YHWH (v.10), who requires judgment for Achan's sin. In chapter 9, we see forgiveness (v.20) in the name of the Lord (v.9, 19) performed to the Gibeonites even in the midst of their having sinned (v.22), for they believed (v.24). Both instances testify to the character of the Person bearing the name of the Lord to forgive those who believe (our point #2). As best I can tell, there is no reference in Joshua to the coming Judge, though Joshua (which means "Savior") himself is appointed by God to lead and judge the people (1:5-9), and it is he who is central to enacting the judgments for both Achan and the Gibeonites. So if one holds to typology, then Joshua is picturing the appointed Judge of God.
In 11:27, it is established that "the Judge" is YHWH. This places at least one direct "name" to the Judge. The whole book of Judges serves as showing human judges being appointed by the Judge to enact judgement.
YHWH is again noted as judge in 1 Sam 2:10 (cf. 24:12, 15), but the verse also bears more weight in the discussion:
The adversaries of the LORD shall be broken in pieces; From heaven He
will thunder against them. The LORD will judge the ends of the earth.
“He will give strength to His king, And exalt the horn of His
This is the first clear reference to some coming king and anointed one, and given in the context such that YHWH's judging implies to be through this individual (though that is not explicit here). Even before Israel asked for a king to judge them (1 Sam 8:6, 20), God already had a king in mind for them.
The king that will rule is found to come from the line of David, who himself was judge of Israel (2 Sam 8:15), but the more importantly the line through which the throne would find its final establishment forever (2 Sam 7:12-13):
12“When your days are fulfilled and you rest with your fathers, I will
set up your seed after you, who will come from your body, and I will
establish his kingdom. 13 He shall build a house for My name, and I
will establish the throne of his kingdom forever.
In 1 Sam 12:10-11, the book of Judges is referenced as a demonstration of God's forgiving of sins when people turn to him.
God's name associated with judgment and forgiveness of sins received to those turning to Him is found throughout Solomon's prayer of 1 Kg 8:29-53.
Pause to summarize: The testimony of the Former Prophets indicates that if Peter is referring to the Former Prophets at all, then he must be referring to them as either a single unit, or possibly a reference to Samuel only since it does contain the relevant points. This leaves in play options 2, 3, 4, or 5 in our possible meanings, since the only one book of the Former group that brings the whole of the pieces together is that of Samuel (or Samuel/Judges). Across the four of them there is further testimony to all the relevant points.
It remains to be determined if the Latter Prophets show the relevant points, at least as a group.
Isaiah has many references to the YHWH judging (2:4, 3:13-14, et. al. ), but a few verses specifically note the coming Judge, His name, and forgiveness through him:
From chapter 7
6 For unto us a Child is born,
Unto us a Son is given;
And the government will be upon His shoulder.
And His name will be called
Wonderful, Counselor, Mighty God,
Everlasting Father, Prince of Peace.
7 Of the increase of His government and peace
There will be no end,
Upon the throne of David and over His kingdom,
To order it and establish it with judgment and justice
From that time forward, even forever.
The zeal of the LORD of hosts will perform this.
Chapter 11 also speaks of an individual given judgement that is from the line of Jesse (i.e. David's line; v.1), who judges (v.3-4). But another key verse bringing into focus the Person is 33:22:
For the LORD is our Judge,
The LORD is our Lawgiver,
The LORD is our King;
He will save us
The Isaiah 9 and 33 both point to a King that is Judge, that is God, the Everlasting Father, and yet is also a Son born to mankind. But then the ch. 33 reference continues and in v.24 notes:
The people who dwell in it [Zion, Jerusalem, v.20, the city where there is favor from the King, v.21] will be forgiven
The connection with the LORD and this city in forgiving sins is found as well in Isa 40:2. But that iniquities forgiveness is received through the human instrument bearing God's name, we find Isa 53:11 (cf. v.5-6) testifying:
He shall see the labor of His soul, and be satisfied.
By His knowledge My righteous Servant shall justify many,
For He shall bear their iniquities.
Justifying many is an act of judgment, while bearing iniquity is not imposing the iniquity upon them. This Servant is earlier revealed in 49:7—
Thus says the LORD,
The Redeemer of Israel, their Holy One,
To Him whom man despises,
To Him whom the nation abhors,
To the Servant of rulers:
“Kings shall see and arise,
Princes also shall worship,
Because of the LORD who is faithful,
The Holy One of Israel;
And He has chosen You.”
This Servant is chosen of God, above kings, performs judgment, bears iniquities, and forgives the iniquities of God's other servant, namely Israel (Isa 44:21-22).
The connections to the Servant, the King, the coming Judge, and His role in forgiveness could be continued, but the point I think is made connecting the points of the Judge to come.
While there are other passages in the book, chapter 33 contains the essential elements spread out in verses, first v.14-15,
14 ‘Behold, the days are coming,’ says the LORD, ‘that I will perform
that good thing which I have promised to the house of Israel and to
the house of Judah:
15 ‘In those days and at that time
I will cause to grow up to David
A Branch of righteousness;
He shall execute judgment and righteousness in the earth.
And the "good thing" of v.14 is, in part, forgiveness of sin (v.8). This forgiveness is found in a new covenant God is to make with Israel (31:33-34).
Similar themes are found in Ezek 37:24-28 and 39:21-24.
Similar themes are also found in Hosea 13:10, Joel 3:2, 12, Micah 7:18, Zech 3:7-15 (and probably others, but I must end somewhere).
The evidence bears out that Peter must be referring to "all the prophets" as either options 2, 3, 4, or 5 suggest in the Nevi'im writings, since the relevant points are not contained within each of the writings of the Former Prophets (though all the relevant features can be found within each of the four books of the Latter Prophets and Samuel).
Further historical evidence is needed to establish when the writings were grouped/categorized into the major divisions of Former/Latter. The only conclusive statement is that #3 or #5 could be accurate: Peter could be focusing on the four books of the Latter Prophets alone, since those divisions are historically evident, and those writers clearly deemed prophets; or including Samuel, as he also is evidently recognized as an actual prophet and the composer of some of the writings contained in that group.
Books from both the Former Prophets and the Latter Prophets testify of what Peter speaks. I tend to lean presently toward Peter's reference being to the Books of Samuel (as the first recognized prophet during the kingdom of Israel) and the four Latter Books as the bodies of writing he refers to.