I’ve already dealt with one aspect of this question, in my paper, Genesis, Joseph, Archaeology, & Biblical Accuracy (+ A Brief Survey of Evidence for “The King’s Highway” in Jordan in the Bronze Age: Prior to 1000 BC) [6-8-21]. Also, my article, Abraham, Warring Kings of Genesis 14, & History (7-31-21), included the following relevant information from the eminent Egyptologist and archaeologist Kenneth A. Kitchen:
Price of Slaves in the Ancient Near East
From ancient Near Eastern sources we know the price of slaves in some detail for a period lasting about 2,000 years, from 2400 B.C. to 400 B.C. Under the Akkad Empire (2371–2191 B.C.), a decent slave fetched 10–15 silver shekels, though the price dropped slightly to 10 shekels during the Third Dynasty of Ur (2113–2006 B.C.). In the second millennium B.C., during the early Babylonian period, the price of slaves rose to about 20 shekels, as we know from the Laws of Hammurabi and documents from Mari and elsewhere from the 19th and 18th centuries B.C. By the 14th and 13th centuries B.C., at Nuzi and Ugarit, the price crept up to 30 shekels and sometimes more. . . .
Joseph is sold to some passing Ishmaelites for 20 silver shekels (Genesis 37:28), the price of a slave in the Near East in about the 18th century B.C. Another reference is in the Sinai Covenant, where Moses, on God’s instructions, sets forth the laws to govern the people when they settle in the Promised Land (Exodus 20 ff.). One of the laws concerns the compensation to be paid to the owner of a slave if someone else’s ox gores the slave to death: The responsible party is to reimburse the slave-owner with “30 shekels of silver” (Exodus 21:32)—reflecting the price of slaves in the 14th or 13th century B.C. . . .
In each case, the Biblical slave price fits the general period to which it relates. If all these figures were invented during the Exile (sixth century B.C.) or in the Persian period by some fiction writer, why isn’t the price for Joseph 90 to 100 shekels, the cost of a slave at the time when that story was supposedly written? And why isn’t the price in Exodus also 90 to 100 shekels? It’s more reasonable to assume that the Biblical data reflect reality in these cases. (“The Patriarchal Age: Myth or History?”, in Hershel Shanks, editor, Biblical Archaeology Review 21:02; March/April 1995)
Currently, I shall draw mostly from the book by archaeologist James K. Hoffmeier: Israel in Egypt: The Evidence for the Authenticity of the Exodus Tradition (New York, Oxford: Oxford University Press, 1996). I provisionally accept the approximate life and death dates for Joseph of c. 1562-1452 BC: arrived at by the Jewish Virtual Library.
To start with, no one is claiming that archaeology provides direct proof of Joseph in Egypt. We haven’t found a plaque that says: “I, Joseph: the Israelite with the coat of many colors, was sold into slavery by my brothers, and am now a big shot in the Egyptian government.” Instead, the argument set forth below is of the following nature; described by Dr. Hoffmeier:
[I]f the narratives look like history, are structured historiographically, and the events described (especially in the Joseph story) are not incredible and compare favorably with the Egyptian backgrounds (as I have shown they do), then the narratives ought to be considered historical until there is evidence to the contrary. . . .
[T]he indirect evidence presented here tends to demonstrate the authenticity of the story. There is really nothing unbelievable or incredible about the narrative. The absence of direct evidence for Joseph, of course, does not disprove his existence because negative evidence proves nothing, while the indirect evidence supports the historicity of the story and its protagonist. (Hoffmeier, ibid., 97)
Scholars with Egyptological training have long recognized the Egyptian elements in the Joseph story. Alan R. Schulman [source] claims the writer of this Hebrew narrative “had an exceedingly intimate knowledge of Egyptian life, literature, and culture, particularly in respect to the Egyptian court, and in fact, may even have lived in Egypt for a time.”
Famine & Drought & the Israeli Refuge in Egypt
Genesis 12:10 (RSV) Now there was a famine in the land. So Abram went down to Egypt to sojourn there, for the famine was severe in the land.
Genesis 26:1-2 Now there was a famine in the land, besides the former famine that was in the days of Abraham. And Isaac went to Gerar, to Abim’elech king of the Philistines.  And the LORD appeared to him, and said, “Do not go down to Egypt; dwell in the land of which I shall tell you.
Genesis 37:28 Then Mid’ianite traders passed by; and they drew Joseph up and lifted him out of the pit, and sold him to the Ish’maelites for twenty shekels of silver; and they took Joseph to Egypt.
Genesis 41:57 Moreover, all the earth came to Egypt to Joseph to buy grain, because the famine was severe over all the earth.
Genesis 42:1-2, 5 When Jacob learned that there was grain in Egypt, he said to his sons, “Why do you look at one another?”  And he said, “Behold, I have heard that there is grain in Egypt; go down and buy grain for us there, that we may live, and not die.” . . .  Thus the sons of Israel came to buy among the others who came, for the famine was in the land of Canaan.
Genesis 43:1-2 Now the famine was severe in the land.  And when they had eaten the grain which they had brought from Egypt, their father said to them, “Go again, buy us a little food.”
Genesis 46:6 They also took their cattle and their goods, which they had gained in the land of Canaan, and came into Egypt, Jacob and all his offspring with him, (cf. 46:32; 47:6 with regard to cattle)
Genesis 47:11-13 Then Joseph settled his father and his brothers, and gave them a possession in the land of Egypt, in the best of the land, in the land of Ram’eses, as Pharaoh had commanded. And Joseph provided his father, his brothers, and all his father’s household with food, according to the number of their dependents.  Now there was no food in all the land; for the famine was very severe, so that the land of Egypt and the land of Canaan languished by reason of the famine.
[T]he East Delta was well fitted for pasturing cattle . . . [this[ is known from the first stela text of King Kamose of Thebes barely 80/100 years later, when his regime could have seasonal grazing for their cattle in the north. (Kitchen, On the Reliability of the Old Testament, Grand Rapids and Cambridge: William B. Eerdmans Publishing Company, 2003, 348)
[E]pigraphic and archaeological data clearly demonstrates that Egypt was frequented by the peoples of the Levant, especially as a result of climatic problems that resulted in drought (as “Merikare” reports), from the end of the Old Kingdom (ca. 2190 B.C.) through the Second Intermediate Period (ca. 1786-1550 B.C.). Even during the Empire period, there are records of hunger and thirst driving people from Canaan and Sinai to Egypt for relief. . . . Thus, for a period roughly from 1800 to 1540 B.C., Egypt was an attractive place for the Semitic-speaking people of western Asia to migrate, and during the final century, Lower Egypt was controlled politically by rulers of Syro-Palestinian origins in Avaris. This span of time coincides with the traditional “Patriarchal Period” and therefore fits the period and circumstances described in Genesis when Abraham, Isaac (almost), and Jacob went to Egypt in search of food, water, and green pastures. (Hoffmeier, 68)
Hoffmeier presents various pieces of additional “indirect evidence” in a chapter from this book entitled “Joseph in Egypt” (pp. 77-106):
Kitchen maintains a late Middle Kingdom [1938-c. 1630 BC] to Second Intermediate Period [c. 1630-1540 BC] setting for Joseph, with the Ramesside period [1292-1075 BC] details indicating the compositional date of the narratives. (Hoffmeier, 79)
I concur with Kitchen that the weight of the Egyptological data, when thoroughly examined, lends both credibility to the essential historicity of the narratives and points to a Late Bronze Age date (i.e., thirteenth century) [i.e., the time of Moses] for the composition of the Hebrew narratives, with possible editorial work being done in the period of Israel’s united monarchy. (Hoffmeier, 98)
Joseph as “Overseer of a House”
Genesis 39:4 So Joseph found favor in his sight and attended him, and he [Potiphar: v. 1] made him overseer of his house and put him in charge of all that he had.
Papyrus Brooklyn 35.1446 [1809-1743 BC] lists the Semitic names of dozens of male and female servants attached to a particular estate. The third column from the right on the papyrus contains their trade or occupation. Interestingly, a number of these servants are identified as hry-pr, literally, “he who is over the house,” which is translated “domestic servant.” (Hoffmeier, 84)
Egyptian names in the Joseph narrative are Potiphar, his master (Gen 39:1), Asenath, his wife (41:45), Potipherah, his father-in-law (41:45), and Zaphenath-paneah: Joseph’s Egyptian name, given to him by the Pharaoh (41:45). Both Hoffmeier (pp. 84-88) and Kitchen (2003, 345-347) go into exhaustive linguistic-historical analysis, not given to brief summary, but his Hoffmeier’s overall conclusion is as follows:
Despite the disagreement among Egypto-Semitic specialists concerning the precise etymology of the four personal names discussed here and their dating . . ., all agree that they are undeniably Egyptian. (Hoffmeier, 87)
That these are Egyptian is beyond doubt . . . (Kitchen, 2003, 345)
Kitchen (2003) believes that the dating of Zaphenath is “Middle Kingdom to early New Kingdom (= early to mid-second millennium)” and Asenath, also Middle Kingdom (p. 346), and Potiphera, 13th century BC (p. 347).
“Pharaoh” vs. “Pharaoh [Name]”
Historians have long lamented that the writer of the Joseph narratives (and those of Exodus, too) did not furnish the reader with the name of the Egyptian king. Throughout Genesis and Exodus, the well-known title “Pharaoh,” . . . is used. [see many scores of examples of this in RSV]. As a reference to the palace, this expression goes back to the Old Kingdom, but as an epithet for the monarch, it does not occur until the Eighteenth Dynasty, sometime before the reign of Thutmose III (1479-1425 B.C.). By the Ramesside period (1300-1100 B.C.), “Pharaoh” is widely used and continued to be popular in the late period. . . . In subsequent periods, the name of the monarch was generally added on. This precise practice is found in the Old Testament . . . after Sheshak (ca. 925 B.C.), the title and name appear together (e.g., Pharaoh Neco, Pharaoh Hophra).
Thus, the usage of “pharaoh” in Genesis and Exodus does accord well with the Egyptian practice from the fifteenth through the tenth centuries. (Hoffmeier, 87-88)
The first appearance of “Pharaoh” in the Bible (RSV) with a name attached is “Pharaoh Neco” in 2 Kings 23:29, 33-35. The dates of the Bible and Egyptian history line up again. Pharaoh Neco killed King Josiah (2 Kgs. 23:29). Encyclopedia Britannica reckons Josiah’s dates as c. 648-609 BC, and Pharaoh Necho II‘s dates or reign as 610-595 BC. Thus, in the second year of his reign he killed King Josiah of Judah.
The only other instance of a Pharaoh being named in the Old Testament is (as alluded to by Hoffmeier), “Pharaoh Hophra” (Jer 44:30). Jeremiah lived in the 7th and 6th centuries.
Genesis 41:41-43 And Pharaoh said to Joseph, “Behold, I have set you over all the land of Egypt.”  Then Pharaoh took his signet ring from his hand and put it on Joseph’s hand, and arrayed him in garments of fine linen, and put a gold chain about his neck;  and he made him to ride in his second chariot; and they cried before him, “Bow the knee!” Thus he set him over all the land of Egypt.
[Donald] Redford . . . [analyzed] over forty scenes spanning from the reign of Thutmose III to the Twenty-first Dynasty (ca. 1479-950 B.C.). These scenes typically show the king sitting on a throne, often under a canopy, while the recipient stands before the monarch wearing a gold necklace and adorned in white linen. . . . An important diagnostic feature of investiture scenes is the presence of some sort of insignia of the new office (standard, staff, or seal). . . .
[A]ll nine of his investiture types range from Thutmose III to Ramesses II. To these Kitchen has added several more, all from the New Kingdom. . . .
The evidence reviewed here strongly suggests that these ceremonies were most popular during the New Kingdom . . . a regular part of the funerary repertoire of the fifteenth through twelfth centuries. (Hoffmeier, 91-92)
Huy, Viceroy of Cush under Tutankhamun . . . [is shown] receiving a rolled-up linen object along with a gold signet ring. (Hoffmeier, 92)
Of the Egyptian nature of the trappings for royal appointments to high office — linen robe, gold collar, state seal, etc. – there can be no doubt whatever. (Kitchen, 2003, 478).
Genesis 50:2-3 reports that Jacob was mummified according to Egyptian prescription, and Joseph himself was placed in a coffin (Gen. 50:26). Neither mummification nor the use of coffins was known in Canaan during the Bronze Age. (Hoffmeier, 95)
High-Level Hebrew in Egyptian Government?
Some have questioned whether a Semite or Israelite / Hebrew could have attained to such a high office in Egypt as Joseph (and Moses) did. But a tomb was discovered in Saqqara (or, Sakkara) in the 1980s that included a Semitic man, Aper-el. His titles included “vizier,” “mayor of the city,” and “judge” and he served Pharaoh Amenhotep III (r. c. 1387-c. 1350 BC) and Pharaoh Akhenaten (r. c. 1350- c. 1335 BC) as overseer or Lower Egypt. Hoffmeier states:
If such a high-ranking official . . . was completely unknown to modern scholarship until the late 1980s, despite the fact that he lived in one of the better documented periods in Egypt’s history, . . . it is wrong to demand, as some have, that direct archaeological evidence for Joseph should be available if he were in fact a historical figure. This point is especially true for Joseph since his provenance in Genesis is the Delta, still an underexcavated area. (Hoffmeier, 94)
How Long Did Joseph Live?
Genesis 50:22 reports Joseph’s age as 110 when he expired. As long ago as 1864 it was recognized that this figure represents the ideal lifetime in Egypt . . . Clearly, the life span of 110 was not an Israelite ideal. Rather, seventy is held up as a normative life span and eighty years as the ideal (cf. Ps. 90:10; 2 Sam. 19:32, 35). . . .
J. M. A. Jansen, in a 1950 article, assembled about thirty occurrences of applications of this age to individuals in Egyptian texts, spanning from the late Old Kingdom down to the Ptolemaic period (ca. two thousand years). Interestingly, the Ramesside period has the greatest number of usages (twenty) . . . (Hoffmeier, 95)
Photo Credit: Joseph Overseer of the Pharaoh’s Granaries (1874), by Lawrence Alma-Tadema (1836-1912) [public domain / Wikimedia Commons]
Summary: Archaeologist James Hoffmeier believes that indirect evidence supports the authenticity and historicity of the biblical story of Joseph in Egypt. I highlight his arguments from “plausibility”.