In 104-103 BCE Judas Aristobulus annexed the Galilee to Judea and forced its Iturean inhabitants, who populated the northern region, to convert to Judaism. ...Judeans probably lived in the Galilee earlier, but it was populated and governed predominantly by the Itureans, the center of whose kingdom was in Chalcis in Lebanon. Their origin is obscure--probably Phoenician and possibly tribal Arab. The territory annexed by Aristobulus stretched from Bet She'an (Scythopolis) in the south to beyond Giscala in the north--that is, most of today's Galilee minus the coast. Masses of Itureans, the original inhabitants of the Galilee, assimilated into the expanding Judean population, and many became devout Jews. One of Herod's associates was Sohemus the Iturean (The Invention of the Jewish People, p. 159).
There are a lot of questions here. My casual research on the Itureans indicates that the limits of their presence in Galilee at the time of Aristobulus is uncertain. The relevant passage from Josephus does not mention Galilee, so I'm not sure where Sand is getting his information--
[Aristobulus] was called a lover of the Grecians; and had conferred many benefits on his own country, and made war against Iturea, and added a great part of it to Judea, and compelled the inhabitants, if they would continue in that country, to be circumcised, and to live according to the Jewish laws. He was naturally a man of candor, and of great modesty, as Strabo bears witness, in the name of Timagenes; who says thus: "This man was a person of candor, and very serviceable to the Jews; for he added a country to them, and obtained a part of the nation of the Itureans for them, and bound them to them by the bond of the circumcision of their genitals" (Antiquities, conclusion to Ch. 13).
That the Itureans were converted to Judaism en masse appears to be well-founded. Many zealous and prominent Jews of the era were converts or descended from converts, a phenomenon I have posted about over at Jewish Texts and History. It seems, then, quite possible that Jesus of Nazareth, if he existed, was Iturean by descent, and if Iturean, then probably Arab.