Occurrences in the New Testament Corpus
ᾅδης (Hades) appears 10 times in the New Testament,1 and the context of each occurrence indicates that it is the abode of the dead.
One particular account references the idiomatic idea of 'Abraham's bosom'2 and includes the idea of a division within Hades where some are comforted and others are tormented in fire, however it is disputed if this account was intended to be taken literally or if it is a reference to an existing Egyptian or Jewish tale.
Occurrences in the Septuagint Corpus
ᾅδης (Hades) occurs 105 times in the Septuagint (LXX, including the Apocrypha).3 Listing the occurrences would be exhaustive, but much can be gleaned by noting what Hebrew words are often translated as ᾅδης in the Septuagint, as depicted in the image below:
4
It is difficult to see in this image, but the same words are actually shown separately (unfortunately) when the word has a prepositional or other prefix in the Hebrew. Combining these, the Hebrew word most frequently (slightly over 56% of occurrences) translated as ᾅδης in the Septuagint is שְׁאוֹל (Sheol).
When used in a locative sense, שְׁאוֹל is best translated from the Hebrew into English as 'wasteland, void, [or] underworld.'5 שְׁאוֹל refers to the abode of the dead in Hebrew thought. It should be kept in mind, however, that ᾅδης is used to translate several other Hebrew terms other than שְׁאוֹל, and ᾅδης occurs 105 times in the LXX corpus as compared to the word שְׁאוֹל occurring only 65 times in the Hebrew Bible corpus (although the Septuagint corpus is larger than the Hebrew Bible corpus as it contains works that do not appear in the latter, not to mention textual discrepancies).
Occurrences in Other Relevant Corpora
ᾅδης (Hades) initially referred to the god of the nether world, and later came to be used primarily in reference to the nether world itself in ancient Greek corpora, i.e. the place of the dead (cf. Arndt, Danker, and Bauer).6 Homer's Iliad gives numerous examples of the earliest usage to refer to Hades as a deity within Greek mythology:
We were three brothers whom Rhea bore to Saturn- Jove, myself, and
Hades who rules the world below. Heaven and earth were divided into
three parts, and each of us was to have an equal share. When we cast
lots, it fell to me to have my dwelling in the sea for evermore; Hades
took the darkness of the realms under the earth, while air and sky and
clouds were the portion that fell to Jove; but earth and great Olympus
are the common property of all.7
Also, since the mythological character Persephone was mentioned in the question, it may be of interest that she and Hades didn't really live 'happily ever after' in Greek mythology; he has to abduct Persephone in order to get her to come to the underworld with him (and likely raped her), which invokes a curse/famine, and the situation keeps getting worse until he is forced to return her, but not without tricking her in such a way that she must return to him annually (due to eating food from the underworld). Hardly a romantic story with a happy ending.
While some additional perspective can be gleaned from an extensive study of ᾅδης in Greek mythology, it is sufficient for the purpose of understanding its use as a metaphysical location (or construct) in first century Jewish and early Christian thought to merely understand that the term initially referred to a mythological deity and then later came to refer to the abode/kingdom under his control, and then this term was conflated with the related Hebrew idea of שְׁאוֹל in the Septuagint and New Testament corpora (much has been written elsewhere on the topic for those desiring to learn more about its earlier mythological understanding).
It should also be noted from study of the corpora that the cosmology of first century Greeks and Jews was significantly different than that of contemporary Western thought (including contemporary Christian cosmological notions of 'heaven', 'hell', and the material universe). It is therefore imperative that modern Western ideologies concerning cosmology and the afterlife not be eisegetically (and anachronistically) read back into the New Testament Greek text.8
Early Christian Interpretation
It has been shown in study of the corpora that the Western notion of 'hell' is very different from the Hellenic understanding of 'Hades,' but at this point the reader may be wondering how the concepts were understood in early Christian interpretation.
Cyril of Alexandria gave a good summary of what was most likely representative of the early Christian interpretations of Hades. In his commentary on John 10:12-13, he writes:
For man, having yielded to an inclination for sin, at once wandered
away from love to God. On this account he was banished from the sacred
and Divine fold, I mean the precincts of Paradise; and having been
weakened by this calamity, he became the prey of really bitter and
implacable wolves, the devil who had beguiled him to sin, and death
which had been germinated from sin. But when Christ was announced as
the Good Shepherd over all, in the struggle with this pair of wild and
terrible beasts, He laid down His life for us. He endured the cross
for our sakes that by death He might destroy death, and was condemned
for our sakes that He might deliver all men from condemnation for sin,
abolishing the tyranny of sin by means of faith, and nailing to His
cross the bond that was against us, as it is written. Accordingly, the
father of sin used to put us in Hades like sheep, delivering us over
to death as our shepherd, according to what is said in the Psalms: but
the really Good Shepherd died for our sakes, that He might take us out
of the dark pit of death and prepare to enfold us among the companies
of heaven, and give unto us mansions above, even with the Father,
instead of dens situate[d] in the depths of the abyss or the recesses of
the sea. Wherefore also He somewhere says to us: Fear not, little
flock; for it is your Father's good pleasure to give you the
kingdom.9
This understanding is consistent with the New Testament corpus usage (particularly elucidating the uses of ᾅδης in Revelation 20:13-14. When read in context (and in tandem with other early Christian writers), the early interpretive idea is that prior to Jesus' death and resurrection, all those who died went to Hades (although there was likely some sort of distinction between the righteous and unrighteous in this 'realm'). However, Jesus descended into Hades after his crucifixion (just like everyone else who died) and 'preached' to the souls there (clearly drawing on the language from 1 Peter 3:19); then he despoiled (from σκυλεύω) it, leading Death/Hades' captives out of Hades (Cyril has recently been falsely accused of universalism because he believed that Jesus led all of the dead out of Hades, not just the righteous ones. But Cyril believed in eternal punishment, he simply didn't link it to 'Hades.' Instead, Hades represented the power of Death, and therefore Jesus 'bound the strong man' and left no prisoners in Hades, thus establishing himself as 'Lord'/κύριος over the living and the dead). Cyril also believed that Jesus united heaven to earth (a common early view).10
Etymology of (and the Highway to) Hell
An etymological dictionary indicates that the word 'hell' has carried the connotation of being an undesirable eternal destination since at least the 14th century.
It is difficult to definitively determine how 'Hades' came to be associated with the concept 'hell.' The most common theory is that the Latin word infernum carried the idea of divine punishment and gradually came to be confused with the distinct concepts of γέεννα (Gehenna), ταρταρόω (Tartaroo/us), and ᾅδης (Hades). The King James Version of the Bible then followed suit, having long forgotten the distinction between the terms, and translated all of them as 'hell.'
Dr. Clark Carlton, in his talk entitled "Hell: A Modest Proposal", espouses his belief that:
...by the time the Scriptures came to be translated into the
vernacular in the medieval west, the Latin word for Hades, Infernus
(or sometimes Inferus) and the transliterated Aramaic word Gehenna had
become completely confused, so that the terms Infernus and Gehenna
were used interchangeably. Initially, Latin authors distinguished
between Infernus and Gehenna just as the Latin translations of the
Bible had done. Augustine, for one, was very careful in his use of
these words. And yet, early on, there was a tendency to import the
notion of punishment into the concept of Infernus. We see this already
in Tertullian, and Cyprian of Carthage actually used the word Gehenna
to refer to the abode of the dead—even before the Last Judgment.
Gregory the Great spoke of the Rich Man being in Gehenna even though
the Vulgate, following the original Greek, uses the word Infernus. By
the time we get to the Venerable Bede, the terms are used
interchangeably—any sense that they refer to different realities
has been lost.
This explains why all English translations of the Scripture prior to
the 20th century, from the very earliest translations of the Psalter
into middle English, to the translations by Wycliffe, and, of course,
the KJV and the Book of Common Prayer, all of them translate both
Hades and Gehenna using the single word Hell. To the translators,
Hades and Gehenna meant the same thing, so they used the most obvious
word in their own tongue to render both terms. The problem is that the
concept of Hell as the state of death, or, more literally, the abode
of the dead, has been completely lost.
Why is Matthew taking a grim look at Hades in Matthew 16:18?
It is clear that the distinction between the Greek terms γέεννα (Gehenna), ταρταρόω (Tartaroo/us), and ᾅδης (Hades) was somehow lost in translation into the Latin Bibles and later into English Bibles. The early Hellenic, Jewish, and Christian understandings of ᾅδης have been explained. It is the recommendation of the author of this post that ᾅδης be transliterated ('Hades') rather than translated (as 'hell'). With that said, this is not intended as a reference to eternal punishment in this passage, but rather as a metonymy for 'the power of death.'11
The IVP New Testament Commentary further supports this idea that ᾅδης is a metonymy for 'the power of death':
The “gates of Hades” in the Old Testament (Job 38:17; Ps 9:13) and
subsequent Jewish tradition referred to the realm and power of death;
death itself would not silence the church. Against those who
presuppose that Jesus could not have planned the church, though he
chose twelve disciples as the nucleus of a remnant for Israel (compare
the symbolic use of twelve in the Dead Sea Scrolls), the language of a
“church” was already being used for a remnant community among his
contemporaries (Dead Sea Scrolls...).12
The translation would thus read,
And I tell you that you are Peter, and on this rock I will build my
church, and the power of death will not prevail against it.
This is a good dynamic-equivalent translation, but a formal-equivalent translation may be desirable (this is a subjective preference). For this reason, it may be preferable to translate it like so:
And I tell you that you are Peter, and on this rock I will build my
ekklēsia, and the gates of Hades will not overpower it.
Jesus was essentially saying, "Nothing can stop us! Not even death!"
The specific location where this occurred is also notable, as I've explained in the "Cosmic Geography" section of a related post.
1 cf. Matthew 11:23; 16:18; Luke 10:15; 16:23; Acts 2:27, 31; Revelation 1:18; 6:8; 20:13, 14.
2 The Greek text reads, "ἐν τοῖς κόλποις αὐτοῦ" (the referent of αὐτοῦ being Ἀβραὰμ in this clause).
3 Based on a search conducted on February 4, 2014 using Logos Bible Software 5.2 SR-4 of Septuaginta: With Morphology. Electronic ed. (Stuttgart: Deutsche Bibelgesellschaft, 1979). Other Septuagint texts were also used, resulting in little notable variation of relevance to this question (108 in Swete's LXX compared to 105 occurrences cited).
4 Analysis performed using Logos Bible Software 5.2 SR-4.
5 Ludwig Koehler et al., The Hebrew and Aramaic Lexicon of the Old Testament (Leiden; New York: E.J. Brill, 1999), 1369.
6 "Orig. proper noun, god of the nether world, ‘Hades’, then the nether world, Hades as place of the dead, Ac 2:27, 31 (Ps 15:10; Eccl 9:10; PGM 1, 179; 16, 8; Philo, Mos. 1, 195; Jos., Bell. 1, 596, Ant. 6, 332). Of Jonah’s fish ἐκ τοῦ κατωτάτου ᾅδου. In the depths, contrasted w. heaven ἕως (τοῦ) ᾅδου Mt 11:23; Lk 10:15 (PsSol 15:10; cp.; Is 14:11, 15); ἐν τῷ ᾅδῃ 16:23; ἐν Ἅιδου ApcPt Rainer. Accessible by gates (but the pl. is also used [e.g. Hom., X., Ael. Aristid. 47, 20 K.=23 p. 450 D.] when only one gate is meant), hence πύλαι ᾅδου (Il. 5, 646; Is 38:10; Wsd 16:13; 3 Macc 5:51; PsSol 16:2.—Lucian, Menipp. 6 the magicians can open τοῦ Ἅιδου τὰς πύλας and conduct people in and out safely) Mt 16:18 (s. on πέτρα 1b and πύλη a); locked ἔχω τὰς κλεῖς τοῦ θανάτου καὶ τοῦ ᾅδου Rv 1:18 (the genitives are either obj. [Ps.-Apollod. 3, 12, 6, 10 Aeacus, the son of Zeus holds the κλεῖς τοῦ Ἅιδου; SEG VIII, 574, 3 (III A.D.) τῷ τὰς κλεῖδας ἔχοντι τῶν καθʼ Ἅιδου (restored)] or possess.; in the latter case death and Hades are personif.; s. 2). ὠδῖνες τοῦ ᾅδου (Ps 17:6) Pol 1:2; Ac 2:24 v.l. (for θανάτου). εἰς ᾅδου (sc. δόμους B-D-F §162, 8; Hom. et al.; Bar 3:11, 19; Tob 3:10; En 102:5; 103:7; Ar. 11, 3) Ac 2:31 v.l.; 1 Cl 4:12; 51:4 (Just., D. 99, 3 ἐν ᾅδου μένειν; Mel., Fgm. 8b, 44 τοῖς ἐν ᾅδου νεκροῖς; Iambl., Vi. Pyth. 30, 179 ἐν ᾅδου κεῖσθαι τὴν κρίσιν; Hierocles 14, 451 τὰ ἐν ᾅδου κολαστήρια; Simplicius in Epict. p. 108, 14 punishments for sinners ἐν ᾅδου)."
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), 19.
7 Homer, trans. Samuel Butler, Iliad, Book 15 (XV), retrieved from the Internet Classics Archive (MIT).
8 Some translators transliterate rather than translate this word, rendering it Hades, thereby reflecting the distinction from the two other Greek words that have been translated hell—γέεννα (a dump that was kept burning outside the city) and ταρταρόω (occurring only in 2 Peter 2:4 within the New Testament corpus).
9 Cyril of Alexandria, trans. P.E. Pusey and T. Randell, Commentary on John, Book 6, Vol. 2 (London: Walter Smith, 1885), 76-77. Retrieved from Roger Pearse, "Early Church Fathers - Additional Texts" at http://www.tertullian.org/fathers/cyril_on_john_06_book6.htm#p76, emphasis mine.
10 cf. Daniel A. Keating, "Christ's despoiling of Hades: according to Cyril of Alexandria." St. Vladimir's Theological Quarterly 55, no. 3, 2011: 253-269. Available from SVS Press.
11 cf. Biblical Studies Press, The NET Bible First Edition; Bible. English. NET Bible; The NET Bible (Biblical Studies Press, 2006), Mt 16:18.
12 Craig S. Keener, The IVP Bible Background Commentary: New Testament (Downers Grove, IL: InterVarsity Press, 1993), Mt 16:18.