Adding my $0.02 as an Egyptologist who specializes in Egypt-Near Eastern interactions.

>No, the case for Israelites in Egypt is very weak.

We should be careful not to conflate “Israelite” with “Jewish,” though, and they asked about Jewish people in Egypt. The more relevant answer is that Judaism did not evolve out of Canaanite polytheism until the Iron Age, several centuries after the purported time of the Exodus. Not only are Jews not attested in New Kingdom Egypt, they are not attested anywhere in the Late Bronze Age. Even the papyri of the community of the Jewish temple at Elephantine, which date to the mid-1st millennium BCE, are in many ways strikingly at odds with, and show a general ignorance of, the historical accounts of the Hebrew Bible/Old Testament and suggest that accounts like the Exodus story had not yet been formalized.

Certainly “Israelites” — by which I mean people from the southern Levant in the region that would later become Israel — are quite well attested in Egypt throughout the Late Bronze Age. Of course, Egyptologists do not refer to these people as Israelites but rather Canaanites or Asiatics, partially because their precise geographic origins are often uncertain and partly becase the term Israel is not attested until the reign of Merneptah, as you noted.

Many Canaanites in Egypt were prisoners of war, brought back to Egypt in the thousands. The royal household in particular was full of servants of foreign extraction, and high-ranking nobles often had foreign servants as well. In a letter to his viceroy of Kush User-setet, for example, the 18th Dynasty king Amenhotep II mentions Near Eastern women in User-setet's household.

>You have taken up residence [in Nubia], a brave one who plunders in all foreign countries and a chariot-warrior who fought for His Majesty, Amenhotep II, who takes tribute from Naharin and decided the fate of the land of Ḫatti, the lord of a woman from Babylon, a maidservant from Byblos, a young maiden from Alalakh, and an old woman from Arapḫa...

Others were members of the elite. It was a standard practice from the reign of Thutmose III onward to raise the children of subject rulers in the Egyptian court as hostages before installing them on their fathers' thrones. This not only forged a bond between the Egyptian and Canaanite princes in the royal nursery (Egyptian kAp) but also instilled Egyptian values in the young Canaanite princes and princesses. This practice was later adopted by the Assyrians, and one sees similar hostages raised in the Neo-Assyrian court (e.g. the Arabian princess Tabua and the Babylonian noble Bel-ibni).

Immigrants in search of greener pastures and political refugees also traveled to Egypt. The most famous example of the latter is not a Canaanite but rather a Hittite, the deposed king Muršili III, who fled to Egypt after his uncle seized the throne in a coup.

Of course, the reverse is also true, and Egyptians often moved or traveled abroad. For example, a man with the Egyptian name of Amenmose (attested in cuneiform as Amanmašu) seems to have worked in the royal court of Ugarit and possibly also Carchemish and owned a cuneiform and Anatolian hieroglyphic seal. I provided more examples in one of my past r/askhistorians posts, Are there any records from pre-Achaemenid Egypt of ethnic Egyptians living and working outside of Egypt for foreign peoples?

>It may be that the story of the Exodus related to the Aegean and Anatolian sea peoples, particularly their battles against Egypt alongside Lybia against Merneptah (the main subject of the Israel Stele) and thereafter, later appropriated by the Israelites after their forced relocation into Israelite areas by Ramses III.

It should be noted that there is nothing even approaching a consensus when it comes to the Exodus account, and this is your own theory in particular.

There are many other interpretations of the Exodus story. One theory, championed by Richard Friedman, is that only a very small subset of the Israelites originated in Egypt (more specifically, the Levites). Others believe it is a garbled memory of the formation of Israel set against the context of the political vacuum resulting from the withdrawal of Egyptian forces from the southern Levant. (I personally find this the most likely.) Still others, such as Jan Assmann, trace the origins of the story not to a specific historical incident but rather mnemohistory, a collective memory through which the Israelites forged a common identity based on past events (regardless of the (un)reliability of the historical narratives).

To be fair though - ancient Egypt is known for expunging their histories of things that made them look bad.

So a lack of historical records itself isn't super surprising.

This isn't evidence that it DID happen either though.