The traditional date of the Hebrew exodus from Egypt is 1250 BC,* which would place it during the reign of Ramesses II, the greatest pharaoh of New Kingdom Egypt. Ramesses II is famous for his lengthy reign of sixty-six years, and for siring nearly a hundred children (between 48 and 50 sons, and 40 to 53 daughters). He is also remembered for his numerous military campaigns both against the Nubians in the south, and the Hittites to the north – and for being a signatory to the world’s first peace treaty.
In another attempt at linking up Biblical with Egyptian history, people have also claimed that the Hebrews settled in the Nile Delta during the second intermediate period under the Hyksos, but then “there arose a new pharaoh who knew not Jacob” – perhaps this was Ahmose of Thebes, who reconquered the Delta around 1550 BC, reunifying Egypt, founding the New Kingdom – and enslaving the Hebrews, who remained in this condition until Moses led them out of Egypt some three hundred years later.
(Then there is Sigmund Freud’s theory that Moses himself was a priest of Atenism, Akhenaten’s failed attempt at introducing a monotheistic religion in the 1340s and -30s. The Hebrews got their idea of one God from the Egyptians!)
Of course, this all doesn’t quite work. New Kingdom Egypt controlled Palestine – the border with the Hittite Empire was established at Kadesh, in what is now northern Lebanon. In fact, Ramesses II himself was still on the throne, if the Hebrews left Egypt around 1250, and spent forty years in the Sinai desert, before beginning their conquest of Canaan under Joshua in 1210. The Bible records no fighting against any Egyptians during this time, however; certainly not against Ramesses II. The Wikipedia article on The Exodus says that Biblical details:
point to a 1st millennium BCE date for the composition of the narrative: Ezion-Geber (one of the Stations of the Exodus), for example, dates to a period between the 8th and 6th centuries BCE with possible further occupation into the 4th century BCE, and those place-names on the Exodus route which have been identified: Goshen, Pithom, Succoth, Ramesses and Kadesh Barnea – point to the geography of the 1st millennium BCE rather than the 2nd.
It is possible that a group of people that later merged into the Hebrew nation had Egyptian origins in the late Bronze Age, and whose story was elaborated and then incorporated into the overall narrative. But apart from some ambiguous references to the Habiru, no non-Biblical evidence has yet come to light about such a group.
* Coincidentally also traditional date of the Fall of Troy.