The Book of Kings (1 Kings 6:1) tells us explicitly when the Biblical Exodus happened. It occurred 480 years prior to the reign of King Solomon. King Solomon roughly ruled around – give or take. 480 years earlier would set the Biblical Exodus around – give or take a few decades.
More precisely, the date of the Exodus figured this way would have been , putting it in the reign of Thutmose III (see below).
During the reign of Ahmose I there is a famous stele called the "storm stele" recording terrible calamities in Egypt, including awful storms and days of darkness.
This doesn't narrow it down much, since sandstorms are regular occurrences in Egypt.
More than this, some have dated the Ipuwer papyrus, which is housed in Leiden, Holland, to Ahmose's reign. The author of the papyrus witnesses to days of darkness, the river turning into blood, and the slaves leaving with the wealth of Egypt.
This is assuming the conclusion. The "dating" of the Ipuwer papyrus to the reign of Ahmose I is on the basis of the mention of invaders, not a bunch of people leaving — thus the reference is probably to the Hyksos invasion. So the logic is that the Hyksos invaded, and then there is the mention of poor people being rich, and blood in the river, so the Exodus was the expulsion of the Hyksos. But the correlation between the Hyksos and the Israelites hasn't been established.
There is archaeological evidence of a destruction layer in Gezer dating to the reign of Amenhotep III ().
A destruction layer over 100 years later isn't even circumstantial evidence.
"In the four hundred and eightieth year after the Israelites came out of Egypt, in the fourth year of Solomon's reign over Israel, in the month of Ziv, the second month, he began to build the temple of the Lord."1 Based on synchronisms with Assyrian and Babylonian accounts, Solomon would have been crowned around the year .2 So the fourth year into his reign would have been , and 480 years prior would be , in the reign of Thutmose III ().3
The problem is that there isn't another shred of evidence in support of Thutmose III being the pharaoh of the Exodus. So many historians have concluded that "480" was just one of those suspiciously round numbers so frequently found in the Bible. Biblical numbers are sometimes allegorical instead of literal. "40" is especially common as a generic quantity. Thus "480" is suspiciously round, as Biblical numbers go, being 12 x 40. The actual tenures of the judges were similarly full of rounded numbers. (See this.)
Manetho spoke of the Hyksos, which Josephus considered to have been the Hebrews, and which we now know to have been expelled from Egypt in . The Seder sets the date of the Exodus at . Halfway between these dates (i.e., (1560 + 1312) / 2) is , which is only 10 years away from . It's possible that the author of 1 Kings had access to records of both events, but like Josephus, suspected that they were the same event, and therefore just decided to split the difference.
Ramesses I (): Surmised by Ahmed Osman to be the Pharaoh of the Exodus.
Ramesses II () Also known as Ramesses the Great, he is the most commonly imagined figure in popular culture, but there is no documentary or archaeological evidence that he had to deal with the Plagues of Egypt or anything similar or that he chased Hebrew slaves fleeing Egypt. Ramesses II's late stela in Beth Shan mentions two conquered peoples who came to "make obeisance to him" in his city of Pi-Ramses but mentions neither the building of the city nor, as some have written, the Israelites or Hapiru. Additionally, the historical Pithom was built in the , during the Saite period.
Archaeologists equate the city of "Pi-Ramses", built by Ramesses II, with the city of "Ramesses", one of the store cities that, according to the Book of Exodus (1:11), the Israelite slaves helped to build in Egypt. Thus he would have been the pharaoh who oppressed the Hebrews.
William Albright and Kenneth A. Kitchen were proponents of this.
None of the Above
Another theory is that the whole thing is pure fiction. There is no archaeological evidence of an Exodus. Furthermore, the Canaanites continued worshiping their gods (especially Ba'al) well past any of the proposed dates of the Exodus. Thus the Canaanites didn't become Jewish until much later. One of the leading theories is that the eventual monotheism was a political compromise in the between those who worshiped Yahweh and those who worshiped Elohim, with the result being one god with two names. This was forced by the advance of the Babylonians, who eventually overran Jerusalem.
It is true that pagan idols have been found at Canaanite settlements, through the period in question. But to contend that Judaism didn't exist at the time, because there is evidence of people still worshiping pagan gods, is not legitimate logic. If we make an archaeological site out of a typical modern community, and we find some statuettes of Mother Mary in some of the houses, do we conclude that Judaism doesn't exist, because clearly some of the people in town are Catholics? Or back to ancient times, we can observe that pagan idolatry has been found at Amarna — does that mean that Atenism hadn't been invented yet?