There is no especially close correspondence between Homer's Proteus, as a Halios Geron, "Old Man of the Sea," and any deity in the Ancient Egyptian pantheons. Two Egyptian water deities could be noted as somewhat, very vaguely approximate: Wadj-wer and Nu[n], who are themselves at rather opposite ends of the spectrum in their own world.
Wadj-wer
Wadj Wer was a fertility god who's name means "Great Green". His
origins were probably either off Egypt's Mediterranean coast or along
the major lagoons of the Nile Delta such as at Lakes Mariut, Idku,
Burullus and Mazala. As early as the Old Kingdom this deity is shown
in a relief from the pyramid site of Abusir. He proceeds among the
fecundity figures, carrying an offering loaf on a mat and with symbols
of life (the "ankh" sign) suspended from his arm. Under his
androgynous form with an emphasized breast and a belly indicative of
pregnancy, Wadj Wer is clearly associated with procreation and
prosperity. Water signs are carved across his body suggesting the rich
fishing in the Delta lakes.
TourEgypt.net's article "Wadj Wer: The Pregnant God"
According to Cow of Gold: An Encyclopedia of Egyptian Mythology, Wadj-wer was "A Hapi-like fecundity figure" who "was long believed to be the personification of the Mediterranean Sea, but is now thought to have actually represented the large lakes and lagoons of the north Delta region."
Wadj-Wer was shown as a man with long hair, a large belly, and
heavy breasts, symbolizing the rich yield of the area he represents.
He was often depicted as carrying the ankh, a palm branch, or trays of
food. His body was always covered in symbolic waves.
Perhaps Wadj-wer's modern identification with the Mediterranean has to do with associating the meaning of his name with the term by which many ancient inhabitants of the Ancient Near East referred to the Mediterranean: the "Great Sea." If Wadj-wer really was a personification of this body of water then he would indeed come quite close to being a counterpart to Proteus, as well as several other major Greek sea-gods who were to some extents themselves personifications of this sea.
However, even if this identification of Wadj-wer were accurate, there is the fact of his androgyny. The reason that the Cow of Gold refers to him as "Hapi-like" is that he bears a striking resemblance to the much less obscure Nile deity Ḥāpi, with whom he shares the attributes of being depicted as a man with overt female attributes, such as being pregnant and apparently lactating.
Although Greek water deities (both of salt-water and fresh) were typically also conceived of as fertility deities, they are invariably represented as unambiguously male, even in their common animal form as bulls. (There are female Greek water deities too, and these, correspondingly, are unambiguously feminine.) The Greek gods of rivers in particular often appeared as bulls. Neilos, the personification and god of the Nile River in Greek myths, arguably does not have much, if anything, to do with the main Egyptian Nile personification Ḥāpi. In the same vein, Wadj-wer is quite unlike Proteus or any of the other Greek marine geriatrics (or any other Hellenic sea-god).
Nu[n]
In Egyptian cosmogony, especially at Heliopolis and Hermopolis, the universe is supposed to have begun as an inert mass of water containing all the matter which later separated to form the different elements and compounds that make up recognisable objects and distinct lifeforms, including the gods. This water-mass was itself a god, named Nu, or Nun, whose properties strikingly parallel the watery abyss in Hebrew and Mesopotamian cosmogonies.
In all these cosmologies the entire universe seems to exist as a bubble inside the endless expanse of primordial fluid. The counterpart to this in Greek myth, rather than Proteus or any other mainstream marine divinity, is found in the somewhat unorthodox cult of the Orphics, who say that one of the first entities to exist was the god Hydros, literally "Water." Together with some other primaeval beings, Hydros would go on to engender the rest of the universe. There is correlation here with the Titan Okeanos [Oceanus], whose cosmic body formed a stream which encircled the entire earth.
In the mainstream version of the Greek cosmogony, Khaos [Chaos], "Gap," represents virtually all the same things that Nu does, except that Khaos is more of an admixture of all the elements in such a way that he (or she, maybe) is described as more of a vast emptiness (hence the name "Gap") rather than a solid or liquid bulk. In this version Khaos is the origin of the material that everything else in existence derived itself from. Okeanos is a generation or two younger than the first beings to emerge from Khaos, and one of his numerous daughters would become the mother-in-law of Proteus.1 Neither Khaos nor Hydros nor Okeanos is a Halios Geron, however (although two of Okeanos' half-brothers are2), even if we could more plausibly connect them to Nu.
The English word "ocean" comes from Okeanos' name. At least one British Egyptologist3 has this to say regarding Nu:
It must also be noted that the ocean and also the Nile were identified
with Nu, whose characteristics changed during the latter part of the
dynastic period. The name of this god has been compared with the
Coptic word "abyss," "deep," and the like, and
it is possible that it may have some connection with it...
Perhaps by "ocean" here this writer could also intend "sea," and so perchance a loose connection could be made therefrom to a Greek sea deity, although that, I think, would be tenuous.
The King and His Origins
Having disclaimed all of that, there is in Greek mythology a rather different version of Proteus, who was a mortal king of Egypt. Like two of his predecessors on the throne,4 he was a son of Poseidon. Through this king's father, the King of the Sea, then, we still have the connection, in the Greek myths, between Egypt, together with Proteus, and the sea. King Proteus, like the sea-god Proteus, is also connected to the Trojan War saga, even more intimately.
The most popular version of his involvement therein is that while on the way to Troy from Lakedaimon, Paris and Helen passed through his domain. Proteus substituted Helen for a phantom version of herself, which he sent Paris off to Troy with. For nine years thereafter, then, the Greeks fought the Trojans essentially for nothing, for an illusion. Menelaus later had to come to Egypt to retrieve the real Helen from thence. Also Proteus' sons Polygonos and Telegonos challenged Herakles to a wrestling-match and both paid for the effort with their lives.
Ancient writers after Homer, just as well as modern ones, have made attempts to expound upon King Proteus and his connection to Egypt and even to the divine marine version of the same character, either via the etymology of his name or by equating him with a particular "historical" Pharaoh. According to Diodorus Siculus:
After the death of this king [“Mendes, whom some call Marros”]
there were no rulers for five generations, and then a man of obscure
origin was chosen king, whom the Egyptians call Ketes [Cetes], but who among
the Greeks is thought to be that Proteus who lived at the time of the
war about Ilion.
Some tradition records that this Proteus was experienced in the
knowledge of the winds and that he would change his body, sometimes
into the form of different animals, sometimes into a tree or fire or
something else, and it so happens that the account which the priests
give of Ketes is in agreement with that tradition.
For, according to the priests, from the close association which
the king constantly maintained with the astrologers, he had gained
experience in such matters, and from a custom which has been passed
down among the kings of Egypt has arisen the myths current among the
Greeks about the way Proteus changed his shape.
For it was a practice among the rulers of Egypt to wear upon
their heads the forepart of a lion, or bull, or snake as symbols of
their rule; at times also trees or fire, and in some cases they even
carried on their heads large bunches of fragrant herbs for incense,
these last serving to enhance their comeliness and at the same time to
fill all other men with fear and religious awe.
Anne Burton has written a commentary on this passage of Diodorus,5 saying that:
The identity of the Egyptian king called by the Greeks Proteus
remains uncertain. Diodorus is the first to attribute to him the
Egyptian name Cetes, a form which appears to bear no relation to the
known name of any Egyptian king. It is, however, very likely that
Cetes is the Greek form of the Egyptian ḥȝty-Ꜥ “chieftain”, or
“local prince”✭, particularly as Diodorus specifically says that this
king was of obscure origin, i.e. he was not a member of the ruling
dynasty.
In Homer Proteus appears as an old man of the sea, but even here
he is called Αίγύπτιος, and is said to live on the island of Pharos.
In Herodotus, II, 112, he appears for the first time as a king, and
therefore mortal, and here he is said to live in Memphis. In
Euripides, Helen, Proteus appears as king, not δαίμων, of Pharos;
while Vergil and Ovid both associate Proteus with Carpathus, an island
between Crete and Rhodes, probably used to indicate roughly the same
area of the Mediterranean as Homer. Elsewhere, Proteus appears to have
been associated with Thrace, and legends portray him as coming from
Thrace to Egypt. Spiegelberg, “Der ägypter König Proteus”, BIFAO,
XXX, 1930, 103-106, suggests that the confusion between the Greek and
Egyptian Proteus arose when the Greeks saw statues of kings with the
attributes of the Nile-god, and related them to their own Proteus. But
this is not entirely convincing, as Proteus appears never to have been
seen as a fruitful river god. Wiedemann, Herodots zweites Buch, p.
432, suggests that the Proteus of Herodotus’ account was not the Greek
figure, but an unidentified Egyptian who was compared with him. This
explanation is made the more plausible by the similarity between the
Greek name Proteus, and the Egyptian title pȝ-rwty.✭✭
A couple of Burton's footnotes being equally informative, I have embedded them into my own, below, q.v.
Burton references a Herodotus commentary by Walter Wybergh How & Joseph Wells,6 which says the following under the heading "Homer and Egyptian history", on Historiai 2.112.
The words, "a man of Memphis," imply that Proteus was of a different
family from the previous kings. Homer (Od. iv 126) makes Polybus
king in Egyptian Thebes at the time of the Trojan War, and Manetho
(FHG ii. 581) identifies him with the last ruler of the nineteenth
dynasty, whom he calls Θούωρις. Diodorus (i. 62) follows H. Perhaps H.
has confused an Egyptian title, Proutî, with the familiar “Proteus.”
It is suggested that, as Proteus is a sea-god in Homer (Od. iv.
385), H. may have identified him with the fish-god (Dagon) of the
Τυρίων στρατόπεδον (112. 2). At any rate, H. does not commit the
absurdity of Diodorus (ut sup.), who explains the famous
transformations of Proteus as a myth due to the Egyptian custom of the
king wearing lion skins and other articles on his head, to inspire
terror and reverence.
Notes
1. The Oceanid Doris, whose daughter, the Nereid Psamathe, married Proteus.
2. The sea-gods Nereus (husband of Doris and father of Psamathe [see prev. note]) and Phorkys [Phorcys], sons of Pontos and Gaia.
3. Budge, E.A.W. 1904. The Gods of the Egyptians, or Studies in Egyptian Mythology. Methuen & Co., London, p. 284.
4. Belos [Belus], son of Poseidon by Epaphos' daughter Libya; and Bousiris [Busiris], son of Poseidon by Epaphos' daughter Lysianassa.
5. Burton, A. 1972. Diodorus Siculus, Book 1: A Commentary. E.J. Brill. Leiden, pp. 182-183.
✭ The main argument against this theory is that the Egyptian ḥ usually disappears in the Greek (cf. Ḥwt-Ḥr—Ἄϑυρ), though it is retained in Coptic as ϧ (Wb., III, 25f.). There is no instance of ḥ becoming κ, which one might expect rather to come from Egyptian ḳ, ẖ or perhaps ḫ. But there is e.g. the anomalous instance of Greek χάμψα from Egyptian msḥ, Coptic [N. 1 on p. 182 in original.]
✭✭ The title pȝ-rwty, “high doors” (of temple or palace), is mentioned by Mallet, Les premiers éstablissments grecs en Égypte, p. 401, n. 3, as “un des nombreux titres donnés aux souverains égyptiens.” … Lauth, Ägyptische Chronologie, p. 181f. was the first to identify this title with the name Proteus. [N. 1 on p. 183 in original.]
6. How, W.W., & Wells, J., 1928. A Commentary on Herodotus. Oxford University Press, Great Britain.