Greek text of Heb. 1:2 according to Textus Receptus (Estienne, 1550):
ἐπ᾽ ἐσχάτων τῶν ἡμερῶν τούτων ἐλάλησεν ἡμῖν ἐν υἱῷ ὃν ἔθηκεν κληρονόμον πάντων δι᾽ οὗ καὶ τοὺς αἰῶνας ἐποίησεν
What is the significance of the word υἱῷ being anarthrous?
Short Answer: It's Significance Leads to some Ambiguity in Focus
There are two pertinent syntactical factors here1
First, Daniel Wallace notes, nouns as objects of prepositions (ἐν here) are a case where regularly definiteness is inherent even though the article is lacking. Here is his statement, but then note what is in the midst of this about Heb 1:2...
So second, Wallace believes the anarthrous use here in Heb 1:2 is to express qualitative purposes. The note he refers to as "mentioned above" regarding Heb 1:2 is this:
His note about "on the continuum" is in reference to an image in his book showing the three ways anarthrous nouns can function, with overlap of qualitative at each end of the spectrum to the other two:
Here is Wallace's statement about what qualitative is:
Wallace seems to not be following his own ideas here. Note the last statement again, "unlike generic nouns, a qualitative noun often has in view one individual rather than the class as a whole," and yet of the Heb 1:2 passage, he stated in the second quote above that "it would be closer to the indefinite than the definite category" (which is then emphasizing class, not individual!).
Further Analysis Leading More to Definiteness
The problem is that as a "class," the individual Christ is one of a kind. That means all the qualitative aspects merge and are expressed only in one individual, and thus point to someone very definite. After all, there is no other "son" δι᾽ οὗ καὶ τοὺς αἰῶνας ἐποίησεν ("through whom also He made the worlds," NKJV), as v.2 itself notes.
So at the very least, Wallace (in my opinion) should have hedged his qualitative argument toward the definite side of the continuum. But I myself would lean toward agreement with the many translations that insert "His" before "son" (KJV, ESV, NASB, NIV, NKJV, etc.), which are correctly making it into a very definite reference (i.e. that the qualitative is not in view as much as the specific individual). The whole context of Heb 1:1-2:8 is leading up to the naming of this son in 2:9, "Jesus," which is a very definite reference. And again, the qualities noted in those verses are only true of this One "son," so the definite identity is not distinguishable into a "class" of individual different from this "specific" individual.
So the conclusion then is that being anarthrous is not really very significant, because clearly there is only one definite individual being noted here.
1 Quotations taken from the discussion in Daniel B. Wallace, Greek Grammar Beyond the Basics - Exegetical Syntax of the New Testament (Zondervan Publishing House and Galaxie Software, 1999), 243-247.
"a son" is in contrast to prophets and angels. The first chapter is "proving" that the title that Jesus received ("son") is great than that of prophets or of angels.
As evidence one can see that Paul asks if an angel ever had God declare that he would adopt an angel:
The rest of the chapter cites various passages showing that the title that Jesus inherited is way higher than the title "messenger" (which is what "angel" means).
The whole chapter is concerned with "proving" that the authority and dignity of a son is greater than that of an angel:
So the anarthrous "son" ("a son") is to argue the loftiness of his being a son as opposed to being a prophet or angel.
In the next chapter he shows that his suffering and humiliation were in line with his being a son because of his mission:
And later on he says that Jesus' example of suffering as a son should hearten all God's children:
There are scholars who believe that the Jewish community to whom "To the Hebrews" was written were believers from a Samaritan background. Samaritans too strong exception to the popular Jewish idea that angels acted as mediators in the giving of the Torah and even carved the tablets:
There may have been strong reservations on their part to have a man act in a mediating role and that may be part of the purpose of the letter, to assuage that concern. Stephen's speech in Acts 6 and 7 is also suspected of showing Stephen to be a Samaritan believer.
I've read somewhere in a commentary that the Samaritans were called "Hebrews" as opposed to Jews. Also in both the Stephen account and in "To the Hebrews" there is apparently a Samaritan view of the temple rather than a Jewish one.