I may have forgotten another name or two, but, generally, those are the only two Homeric names commonly used in English. While I understand that they are two major characters, we would almost never hear of an Achilles, Odysseus, or Agamemnon as a given name in the Anglophone world. What made these two names stay in common usage while others have not?
Again, only asking with regards to their use as English given names. I would assume that in other languages different names may still be used.
A related answer by /u/Iphikrates suggests that names of heroes and Gods weren't commonly used as names for children in Ancient Greece to begin with.
As for Anglicized versions of Ancient Greek names showing up in literature, perhaps this list could tide you over while you wait for an answer from an expert.
The vast majority of the names that are used in the Iliad have never been widely used in English, but a handful of these names have passed into English through various channels, mostly not through the Iliad itself. Of all the names of characters in the Iliad, the two that have been in continuous use as names for people in English the longest are Alexander and Helen, which passed from Greek into Latin and from Latin into English very early due to both of these names having been held by particularly famous and revered ancient figures. The names Hector and Cassandra first passed into English a bit later via the medieval "Matter of Rome" (i.e., the corpus of romances based on ancient Greek and Roman stories), but they didn't become popular until the eighteenth century.
I will examine the history of each of these names in detail below.
The name Alexander
In the Iliad, Paris, the son of Priam and brother of Hector, has two different names: Πάρις (Páris), which is most likely a Greek rendering of the Luwian name Pari-zitis, and Ἀλέξανδρος (Aléxandros), which is of Greek derivation. The name Ἀλέξανδρος was not especially common in the Greek world during the Archaic Period (lasted c. 800 – c. 490 BCE) or the Classical Period (lasted c. 490 – c. 323 BCE), but it was importantly a name that was used in the royal household of the Argead Dynasty of the kingdom of Makedonia in northern Greece.
Then, at the very end of the Classical Period, King Alexandros III of Makedonia (i.e., the man commonly known in English as Alexander the Great) ended up conquering the Achaemenid Persian Empire and becoming the most famous conqueror in all of Greek history. As a result of this, the name Ἀλέξανδρος saw an explosion of popularity in the Hellenistic Period (lasted c. 323 – c. 30 BCE) and afterward.
The Lexicon of Greek Personal Names lists Ἀλέξανδρος as the sixth most common masculine name attested in Attike, the second most common masculine name attested in the Aegean Islands, Kypros, and Kyrenaïka, the second most common masculine name attested in the Peloponnese, western Greece, Sicily, and Magna Graecia, and the fifth most common masculine name in coastal Asia Minor from Pontos to Ionia.
The name Ἀλέξανδρος subsequently passed into Latin as Alexander and, eventually, from Latin, into English. Alexander the Great was a well-known figure throughout Europe during the Middle Ages, due in part large part to the Alexander Romance, a fictional romance very loosely based on Alexander's historical life that circulated widely during the Middle Ages through a variety of recensions and translations.
The Alexander Romance may incorporate material from sources written as early as the third century BCE, but the version of the text that serves as the basis for the surviving recensions was compiled by an anonymous Greek-speaking author who lived in Alexandria, probably at some point between c. 140 and c. 340 CE. Later authors substantially revised, expanded, and redacted various parts of the text in various ways, resulting in many different versions of it that are sometimes drastically different from each other. Through these different versions, it became one of the most widely read works of secular literature throughout the medieval period.
The name Alexander was already prominent in Britain by the twelfth century CE. Three medieval Scottish kings—Alexander I (lived c. 1078 – 1124 CE), Alexander II (lived 1198 – 1249 CE), and Alexander III (lived 1241 – 1286)—bore the name, as well as the prominent English Scholastic theologian and Franciscan friar Alexander of Hales (lived 1185 – 1245 CE).
The French poet Jacques de Longuyon in his Les Voeux du Paon (written in 1312) names Alexander the Great as one of the "Nine Worthies," a canonical set of nine legendary and historic men known for their extraordinary valor. The canon neatly consists of three "pagans," three Jews, and three Christians, with Alexander being one of the three "pagans."
After Jacques de Longuyon's poem, references to the Nine Worthies became common in literature and artistic depictions of all nine men as a set started showing up as decorations everywhere for the next several centuries, including in city halls, on fountains, in manuscripts, in woodcut illustrations, and just about everywhere you can imagine.
Alexander the Great's fame and, as a result, his name became ubiquitous. Geoffrey Chaucer (lived early 1340s – 1400) has the monk say in "The Monk's Tale," lines 2631–2633:
"The storie of Alisaundre is so commune
that every wight that hath discrecioun
hath herd somwhat or al of his fortune."
Thus, although its popularity has waxed and waned, the name Alexander has remained in continuous use in the English-speaking world since at least the twelfth century CE. The Dictionary of Medieval Names from European Sources has an entry for the name Alexander that lists what looks like over a hundred some sources from medieval England alone from the twelfth century CE onwards, written in Latin, Dutch, Early Modern English, and Walloon, that mention medieval English people named Alexander.
(This answer is continued in the comment directly below.)
Looking at only the greek names in the Illiad, only 4 have ever been in the top 1000 social security names: Pheonix, Achilles, Helen, and Nestor. This shows that other Illiad names have survived to some extent. Interestingly, this does not line up with google books references at all. So while literary use may be the cause of the names' survival, is not the cause of their popularity.
The literature tells us that names have some trends, like the rise and fall of similar-sounding names with each other, but the history of each name is unique, so let's look at these four to spot trends.
Pheonix (SSN rank 238) seems to be part of the trend of naming people after places (Orlando, Dakota, Austin), and not just a callback to the obscure hero.
Achilles (855) and Nestor (~946) do seem to reference the actual heroes, but, as you point out, they are not actually very popular. In fact, Nestor fell off the list in 2007.
Helen (424) seems to have survived because St. Helena was important in the early church and became a common Christian name.
As popularized in Freakonomics, Levitt's research argues that new names are invented or popularised by affluent or well-educated families, and the names climb down the socio-economic ladder over time. If this is true, Illiad names may become more popular (reading the Illiad is a pretty posh pursuit). Illiad names themselves, though, point more towards Hahn & Bentley's argument that "a few elements inevitably become highly popular, even if they have no intrinsic superiority over alternatives."
In general, it is difficult to tell what creates a popular name. It is a combination of how it sounds, economic, social, and political signaling, pop culture, traditional culture, and more. According to Barucca, with the collapse of centralized media, there has been a shift away from the traditional methods of naming. Perhaps this shift will give Illiad-based names a chance to make a comeback.
Sources
https://www.ssa.gov/cgi-bin/babyname.cgi
https://books.google.com/ngrams/
https://journals.sagepub.com/doi/full/10.1177/0956797612443371
https://en.wikipedia.org/wiki/Helen_(given_name)
Levitt, S. D., Dubner, S. J. (2009). Freakonomics: A Rogue Economist Explores the Hidden Side of Everything. United States: HarperCollins Publishers.
https://doi.org/10.1098/rsbl.2003.0045