Is it their conversion date, the Hebrew date on which they were biologically born, or something else? (For questions of halachic age as well as mystical significance)
Although a Ger is considered to be “born anew” when becoming Jewish, this is a halachic concept that relates to family relationship and the like, and it is not a “biological fact.”
The Ger’s birthday thus remains the date that he was physically born on.
Note that there is no actual halachic significance to a person’s birthday, other than determining his age, which is important only for a bar or bat-mitzvah. For this purpose the date of birth is of course the “birthday.”
Interestingly enough, they seem to write in a different answer that the main birthday is when the ger receives his neshama:
Q: When does a ger celebrating his Jewish birthday – on the Jewish date of his birth or on the Jewish date of his giyur? A: He can celebrate both. One day to thank H-shem for giving him his neshoma, and this is surely the main birthday, however he can also celebrate on the day that H-shem gave him his physical body.