The Hyksos were expulsed before 1530 BCE, by the beginning of the 18th Dynasty of the New Kingdom (beginning with its dynasty founder Ahmose, reign 1539-1514 BCE). This is too early to be the Hebrew Exodus.
In the Exodus, it speak of the time when the Egyptians made the Hebrews into slaves, to build the cities of Pithom and Rameses that is within the Goshen region, where Jacob's family had settled.
There were no pharaohs named Rameses until the 19th Dynasty (as well as in the 20th Dynasty), and the 19th Dynasty, which didn't begin until 1292 BCE.
According to my calculation of the Genesis' patriarchs (see
Timeline of the Patriarchs in the 2nd table), Jacob and his family didn't turn up in Egypt until around 1522 BCE, and the famine ended in 1516. Jacob died in 1505 and Joseph in 1451.
There were no reported Biblical 7-Year's Famine between 1524 and 1516 in the historical Egypt, which smack dead in Ahmose's reign (1539-1512). Egypt was still divided and Hyksos still existed and in power in the region of Avaris. Ahmose didn't attempt to drive out the Hyksos, until his 10th year, which means around 1529 BCE, and the Hyksos' dynasty didn't ended and expulsed until 1523 BCE. There is also no report that his advisor was Hyksos, let alone of Hebrew origin; this new king was seeking an independent Egypt, and reunified Upper and Lower Egypt, so Ahmose would not even consider having any foreigner to be his governor or advisor.
The Hyksos (1630-1523 BCE) were said to rule after the collapse of the Middle Kingdom dynasties in a period known as the Second Intermediate Period, but they existed with the 17th Dynasty (1630-1540 BCE), which were confined in the Upper Egypt, with their capital Thebes. The Hyksos' Dynasty also didn't have any king by the name of Rameses.
I don't think there is any link between the Hyksos and the Biblical Hebrews, except that the Hyksos were known (mistakenly) as the Shepherd Kings, and when Jacob's family came into Egypt as shepherds (Genesis 46:28-34), and they both lived in the Eastern part of the Delta. There is only 5 generations between Jacob and Moses (4 generations, if you look at from Moses' mother's side, since Amram married his aunt, Jochebed, (Amram's father's sister, Exodus 6:20), so the timing is all wrong for the Hyksos-Hebrew connection.
Jacob had only arrived in Egypt, just before the Hyksos dynasty had fallen, and there is too much time between Hyksos' expulsion to coincide with Israelites going on Exodus with Moses, who supposedly left during the time of Rameses' Dynasty.
I think it would make it a lot easier if the compilers of the Genesis and Exodus, would actually put names of the Egyptian kings, ruling during the time of seven-year's famine and of Moses' time.
I don't trust Josephus' supposed source of Manetho, or even Manetho himself is accurate in any way.