The only time Joseph is actually mentioned in an Egyptian text is the "Osarseph" report in the "Aegyptopaea", the Egyptian history of the Egyptian/Greek historian Manetho, thought to be written on the basis of Egyptian temple documents and by direction of king Ptolemy II of Egypt in Memphis around 300 B.C. In this report, Osarseph, a renegade priest of Heliopolis of Asian or Hyksos origin, instigated an uprising of Asian or Hyksos slaves, who were remnants of a once large Hyksos population driven out of Egypt centuries earlier (by the pharaoh Ahmose in 1521 B.C. as we know from his victory stela today), and who were considered "unclean" or "leprous" by the Egyptians, probably meaning that they were corrupted by an un-Egyptian faith, and therefore banned to work in the Egypt-wide stone quarries first, and then locked away in the town "Avaris", the former Hyksos capital in the north-eastern delta of the Nile.
In Avaris, Osarseph renamed himself as "Moses" (which basically means the one who came out of the water as a source of divine inspiration), gave the slaves laws completely contrary to the Egyptian laws (which basically means that he denounced that pharaoh is the supreme god on Earth and hence the supreme god is beyond this Earth and no image or human representative can be made of him), invited contingents of the formerly chased away Hyksos, who had settled down in and around Jerusalem in Palestine, back to Avaris to join the uprising, and launched a religious war on Egypt. In this war, which lasted thirteen years and penetrated Egypt up to Memphis in the south in Middle Egypt, the temples of Egypt were the major targets, and they were deprived of their idols and their sacred animals were being killed. The same war also brought plagues and famine over Egypt.
In the end, the Egyptian pharaoh was able to counter the slave revolt with the help of Nubian reinforcements, and he drove the slaves with Hyksos origin out of Avaris and pursued them right into Syria in the north. These slaves found refuge in Jerusalem and the hill country surrounding this city, and Manetho adds that these slaves are considered as the ancestors of the Jews now.
Two versions of this Osarseph report have survived, both of which quoted in the work "Against Apion" by the first century Jewish historian Flavius Josephus, who was an eye witness of the Jewish war that ended with the destruction of Jerusalem and the second temple in 70 A.D., who became a protegee of the later Roman emperors Vespasian and Titus, who waged this Jewish war, and who wrote extensively about Jewish history in Rome up to his death around 100 A.D. The second version of this Osarseph report deviates from the other version in the detail that it says that Joseph and Osarseph were the leaders of the slave revolt.
And this is the point where the biblical Joseph is actually mentioned in the Egyptian writings. This makes perfect sense, because Joseph according to the Bible was the first of the tribunal faction of the Hyksos migration into Egypt that later became the Israelites to enter Egypt, and Osarseph alias Moses was the one who finally led these people out of Egypt. Also a linkage between Joseph and Osarseph becomes apparent, since Joseph according to the Bible married the daughter of the high priest of Heliopolis. It is therefore very plausible that some of Joseph's offspring remained in the priestly service at Heliopolis, and Osarseph was the last one in that line. One can further observe, since "Osarseph" is obviously a combination of "Osiris" and "Joseph", that there is a direct name relation between Moses and Joseph, and because Osiris is the supreme Egyptian god of creation and the one who presides over the afterlife, one can get a clue from where Moses draw his monotheistic convictions.
Manetho's Orsarseph report has caused much confusion among scholars, starting with Josephus, mainly because the pharaoh mentioned in that report is "Amenophis", a pharaoh Josephus cannot align with any pharaoh in the otherwise conclusive list of Egyptian kings in Manetho's history. Modern scholars mostly identify this pharaoh with Amenophis IV or pharaoh Akhenaten, the "heretical" king who introduced the cult of the single god of the life-giving Sun Aton and also fought a religious war about this issue during his whole reign between about the years 1353 - 1336 B.C. Yet this king makes no sense in a Hyksos setup, and so the scholars conclude that something is mixed up in Manetho's report, maybe based on vague folks traditions, thus leaving the report little credibility.
However, seven different copies of Manetho's kings list have survived, five of which state "Amenophis" for the pharaoh in question, and two state "Merenptah" instead, two pharaohs whose regnal Egyptian names only differ with a syllable. This makes it almost certain that the substitution of "Amenophis" in Josephus' copy is the consequence of an ancient transcription error. Further evidence for this follows from the fact that Manetho's Osarseph report tells us that the alleged pharaoh "Amenophis" named his son after his father "Rameses" and his grandfather "Sethos", but according to the correct and Egyptologically underpinned succession of the kings, it was pharaoh Merenptah who was the son of Rameses II and the grandson of Sethos I (^*). With pharaoh Merenptah instead of "Amenophis" there is no contradiction with the archaeological records. This means that pharaoh Merenptah was the pharaoh of the suppression (probably when he still ruled together with his father) and the exodus of the Israelites.
Pharaoh Merenptah reigned for seventeen years from 1213 B.C. onwards. From this pharaoh a victory stela is preserved, dated for the fifth year of his sole reign, which is the year 1208 B.C., on which he proudly pronounced that he has driven the enemies of Egypt out of the country, namely the Lybians back to Lybia in the north-west, and four other groups back to Syria in the north. One of the latter groups are the "beaten unsettled tribes of the Israelites, whose seeds are no more" (meaning that many of their young warriors have been killed). This Merenptah inscription corresponds to Manetho's report of the pursuit of the Hyksos decendant slaves into Syria or Palestine.
(^*) A reference for this important point is: R. Kittel, "A History of the Hebrews: In Two Volumes", page 260:
... But if it [Manetho's account] does [embody an independent Egyptian reminiscence of the Exodus], the question must then be asked as to what period of Egyptian history it is to be assigned. The names Rameses, Amenophis, Sethos-Rameses (^1) appear to correspond most nearly with those of the Kings Rameses II., Merenptah, and Seti II., who stood to each other in the relation of father, son, and grandson. Accordingly most moderns have agreed that Rameses II. was the Pharaoh of the Oppression [in his later time with Merenptah as a co-ruler] and Merenptah of the Exodus. And, as a matter of fact, it is impossible to deny that by the Amenophis of Josephus, Manetho cannot have meant any one but Merenptah (^2).
(^1) More precisely, in Josephus: Rhampses, Amenophis, Sethos-Rhamesses; in Julius Africanus and Syncellus [versions of Manetho's report]: Rhapeakes (Rhampses), Amenephibes (Merenpthah), Rhamesses. See Lepalus, K"onigsbuch Ant., p. 16f; Ebers, Gosen, p.536.
(^2) The proof is that Julius Africanus and Syncellus actually give [A)menephthes in place of Amenophis. Amenophis must therefore be due to a misunderstanding of the part of Josephus, or an ancient clerical error in his copy of Manetho.