UPDATE (as of May 28, 2012): Anything I have written about Bethel or ‘Ai before late 2011/Early 2012 is ABR-inspired bunk. Read something from this blog’s later months for my present views.
Bethel, as we have seen, was during the Biblical period at el-Bireh and at Ein el Q’asa during the Byzantine period. Beitin (a name without a meaning in Arabic) seems to preserve not so much Beth-aven/Bethaun/Bethar as be a purposeful corruption of Bethel. It is therefore probable that the name “Bethel” was purposefully moved to Ein el Q’asa (or, less likely, el-Q’ada) after one of the Jewish Revolts when el-Bireh was established a Roman military camp (and got its name? Berea?) and, sometime after Eusebius, was moved to Bethar. Ai’s location at Khirbet Maqatir, however, was never forgotten. A Byzantine monastery was built at 31°54’58″N, 35°14’54″E to commemorate Ai, the tent of Abram, or both (NETS: “And from there he withdrew to the mountain east of Baithel and set up his tent there-Baithel towards the sea and Haggai to the east”). However, no Byzantine church or even pot sherds were found at et-Tell.