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.
480 years taken literally, and figured more precisely, would have the Exodus occurring in , putting it over 100 years after Ahmose I came to power (i.e., give or take a century).
Another interpretation of the "480 years" is that the scribes were referring to 12 generations, at 40 years per generation, which is the literary standard in the Tanakh. For example, 40 years shows up several times in the tenures of the judges. (See this.) A more typical average age for men to be having children is 30 years, in which case the total time would have actually been 360 years. Counting back from , this takes us to . In the present thesis, there was an exile of the high priests from Amarna in , and then a somewhat more substantial flight from oppression under Horemheb in .
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.
The Ipuwer papyrus isn't an account of turmoil associated with people leaving — rather, it's an account of the downfall of the existing regime at the hands of people who were arriving. So it might be a description of the invasion of the Hyksos in , but it isn't an account of the expulsion of the Hyksos by Ahmose I in .
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
See the comment above about the number "480" possibly being read more accurately as 360.
Another interpretation of the "480 years" is that the scribes simply split the difference between the exile of the Hyksos in and the exile of the Atenists in . ((1560 + 1312) / 2) = , 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 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.
In the present thesis, there was indeed a migration of the clan of Jacob (i.e., the Israelites) to Egypt during this period, though there was also a settlement of Hebrews in Canaan at precisely the same time. So there were two different groups.
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?