Son of Aigaios?
As far as English translations of the Iliad go, there is a rather unique interpretation of the word Αἰγαίων᾽ in Richmond Alexander Lattimore's 1951 translation of the Iliad (specifically Book 1, Line 404 in this case). A straight transliteration thereof would give us "Aigaion". English translations tend to then Latinise this into "Aegaeon" (such as it is in your Question). In Lattimore's version:
that creature the gods name Briareos, but all men Aigaios' son, but he
is far greater in strength than his father.
This is actually a somewhat ingenious and quite legitimate way to interpret the giant's "human" name. Compare how, in the line immediately following, Zeus is referred to by one of his common epithets, Kronion [Cronion], which, as we know, indicates "Son of Kronos [Cronus]".
On his Poetry in Translation website, A.S. Kline translates the last clause of Line 404 to read that "Aegaeon" is "mightier than his father Poseidon". The word "Poseidon" actually does not appear in this line of the Greek text from which it is being translated, nor do I know of any ancient source that explicitly says anything about this giant being a son of Poseidon.
In his book "The Iliad, Edited with Apparatus Criticus, Prolegomena, Notes, and Appendices" (1900), Walter Leaf says, in Footnote 403 on Page 32, that:
The father of Briareus was, according to the legend, Poseidon, who was
himself sometimes called Αἰγαίων or Αἰγαῖος.
The reference is vague as to the source of this "legend" but there is indeed the citation by Strabo, in Book 9 of his Geography, that Poseidon was called Aigaios at Aigai [Aegae] in Euboia [Euboea], where he had a temple under this name, which presumably denotes him as Poseidon "Of [the City of] Aigai". Book 13 of the Iliad says that Poseidon lived in a palace under the Sea at this Aigai (see also Odyssey 5.381). Lycophron does refer to Poseidon as Aigaion in his poem Alexandra (135).
According to Hesiod's Theogony, Briareus was married to Poseidon's daughter Kymopoleia [Cymopoleia], so at the very least this giant was a son-in-law of the King of the Sea.
There is a lost epic called the Titanomachy which seems to have treated some of the characters from the Theogony differently from Hesiod's placement of their allegiances in his account of the War of the Titans. Referring to this lost Titanomachy, the scholion on Apollonius Rhodius' Argonautica 1.1165, says that Aigaion was a son of Pontos (the Sea) and Gaia (the Earth), that he lived in the Sea, and that he fought on the side of the Titans against Zeus.
Cinaithon, another early poet who knew of this giant's double name,1 says that Aigaion was defeated by Poseidon.2 In some instances it seems that Aigaion and Briareos are not necessarily the same character and that perhaps two separate personages were fused into this one, who we eventually know best as a son of the Sky (Ouranos [Uranus]) and the Earth, rather than of the Sea and the Earth.
It may also be worth noting the possibility that Homer is unaware of a group of triplet giants called the Hekatonkheires [Hecatoncheires]. He never mentions any other apart from Briareos-Aigaion. The Theogony, which does group this giant with triplet brothers, never uses the word Hekatonkheires anywhere and simply calls the most famous one Briareos (or Obriareos). Here, from among these triplets, Poseidon specifically singles Briareos out in order to make him his son-in-law "because he was good" or "goodly." In some sense or another there was something special about him among the fifty-headed divine giants.
The Language Distinction Between the Gods and Mortals
The same sort of double name ascribed to Aigaion-Briareos is given to the main river of Troy in Iliad 20. This river was a god whose fellow deities called him Xanthos [Xanthus] but whom mortals referred to as Skamandros [Scamander]. The same footnote of Leaf's Iliad book, mentioned above, takes note of these double names and comments:
The natural supposition would be that the 'divine' words are archaic
survivals, perhaps from an older race. It is sometimes said that the
divine name has usually a clearer meaning than the human, and that the
Greeks therefore regarded their own tongue as divine, and others as
the languages of mere men.
Leaf, though, goes on to say that this is the case only with one other example, as well as, perhaps the Xanthos-Skamandros Iliad quote, "which, however, look like different renderings of the same foreign word." He goes on to mention that the Pelasgian Hermes was called Imbros [Imbrus], apparently understanding this to be an earlier, pre-Greek version of this deity.
The Roman poet Ovid, in Book 11 of his Metamorphoses, has a similar thing to say about one of the numerous minor gods who personify dreams.
The gods have named him Icelus; here below the tribe of mortals call
It seems clear that Ovid does not mean to say that Latin is the language of the gods.
As with Xanthos/Skamandros and Aigaion/Briareos, this is rather a case, I think, of the poets in question attempting to add verisimilitude and realism to their storytelling. I interpret the idea here to be that the gods have their own language that they speak and that: "Now I, as the narrator, am going to mention a couple of naming conventions as examples of how that divine language differs from that of us mortals [a form of Ancient Greek in Homer's case; Latin in Ovid's case]."
If this assessment is correct then it means that each deity has his own divine name as well as a "human" one. It would, however, detract from the main point, i.e. moving the story along, to spend time revealing (or rather inventing?) an extra fantasy-fiction-style set of names for more than one or two of them. (These works are generally otherwise chock-full of cult titles and special epithets for the gods, and the etymologies of these are almost always something less otherworldly than them being part of a divine language.3)
1. Page 107, from Ch. 8, "Writing Local History: Archemachus and His Euboika", by Sławomir Sprawski, in The Children of Herodotus: Greek and Roman Historiography and Related Genres, ed. Jakub Pigoń (2008).
2. Page 69 of Early Greek Mythography, Vol. 2: Commentary, by Robert L. Fowler (2013).
3. The Theogony (Lines 820ff) implies that the gods do have their own language, which the immense, many-headed, fire-breathing monster Typhoeus has the ability to speak, in addition to emitting " every sort of horrible sound" such as "the sound of a bellowing bull... or again like a lion shameless in cruelty, or again it was like the barking of dogs, a wonder to listen to, or again he would whistle so the tall mountains re-echoed to it."