Hyksos Pharaoh Maaibre Sheshi is considered to be the first king of the 15th (Hyksos) dynasty and is also generally equated to Salitis/Saitis, who is named by Manetho as the first king of that dynasty. I found no reason to challenge the status quo on this point. However, in my model Sheshi/Salitis equates to Babylonian king Abi-eshuuh and Biblical Gershon/Joshua (Patriarch Reu) son of Nun (Patriarch Peleg) and not Abraham. Joshua was named successor of Moses (a.k.a. Eber and Hammurabi) after the disgrace of Eliezer/Joktan (Samsu-iluna).
The following pharaoh, Yakhub-Hor, was the adopted son of Salitis but the true son of Joktan. The two lines are combined in Yakhub-Hor, who is given the Patriarchal name of Serug ("twisted together"). He is succeeded by Khyan (Patriarch Nahor). Khyan appointed Apophis (Terah), but later revoked that election. Apophis defied his father and ultimately reclaimed the kingship. Apophis II (Mamre/Gideon) and Apophis III (Jether son of Gideon) are a son and grandson of Apophis I, respectively. The final Hyksos king is considered to be one Khamudi, who made his final and apparently unsuccessful stand at a place called Tel el-Ajjul. He seems to correspond best to Kamose (Iscah/Saul).
I would also look for Abraham at this time in the late Hyksos period rather than the beginning. There are a number of other Hyksos king names that are known. One of these might belong to our Abraham. For example, at Tel el-Dab'a the Hyksos prince Yansas (Manetho's Iannas) claimed to be the eldest son (heir apparent) of Khyan. The name Yansas/Iannas is perhaps an early form of Jochan/Johann/John, and identifies him as something of an Osiris figure. The co-founder of the Hyksos dynasty was also very clearly an Osiris figure, and therefore the Hyksos dyansty can be considered as something of an Osiride dynasty. The name Joktan ("diminished") makes a ready word play with Jochan ("favored"). We also know that Abraham was called "the Osiris" at Karnak under his Egyptian name of Djehuty.
Other Hyksos king names known from scarabs are: Yat, Aper-Anat, Anat-Har, User-Anat, and Samkuna. A much later genealogy composed for the priests of Memphis makes a king named Sharek roughly contemporary with Apophis I & II. Sharek is perhaps a transposition of Sokar, the epithet of Osiris. The name might also relate to the Hyksos stronghold of Sharuhen just south of Gaza. Interestingly enough, Abraham as an Osiris figure was recognized (at least in the Torah) as founder of the Egyptian New Kingdom. However, unlike Joktan, Abraham was not the true father of the ultimate successor, but only the legal father.
So we have a few candidates for the Hyksos idenitity of Abraham, but not quite enough information to be certain.
See "Egypt, Canaan, and Israel in Ancient Times" by Donald Redford (pp 107-122) for the sources mentioned above.
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