Despite Ibn Ezra and some other commentators, it is unlikely that the word means, “Bow down!” and is connected with berech, a knee. More probably it is a noun and has the connotation of a national title of honour.
The Targum, like Rashi, follows many of the rabbinic sages in interpreting the word as “father of the king”. Av (whether here or elsewhere) does not necessarily mean “father” in a literal sense – in the Bible it is often a word for a captain or chieftain. That this is a valid way of translating the word is seen from Gen. 45:8, where Joseph says that the king has made him an av. In the Talmud (B.B. 4a), rach is taken to mean “king”.
The rabbis suggested that the connection of av and rach (the latter word is “weak” in Hebrew) is that Joseph was rach in years but an av in status.
If we regard the word as Egyptian, it seems to be connected with another Semitic word, avarachu, a title for a high and powerful official.
In yeshivah life, an avrech is a Talmudic student, young in years but senior in wisdom. (In modern Israel some avrechim seem to regard themselves as entitled to bow out of national service.)