I saw in a popular book a suggestion that Avraham was born on Rosh Chodesh Tevet. Does anyone have any backing sources for this?
This page (it is too much to simply copy and paste) speaks of the medrashic concept that the righteous die on their birthdays [scroll down to citation 27 in the text]. As such, acknowledging that Avraham Avinu was righteous, and the gemara mentions possibilities for the Yahrzeit:
Avraham Avinu (2124 - 1638 B.C.E.:), according to some sources (Bava Basra 91a). [According to others, Avrohom Avinu's Yartzeit is 1 Tishrei, or Nissan [Moed Katan 28a].
one could work backwards and derive a birthday commensurate with the death date one accepts.
This answer deals with the sources for the yahrzeit as well.