I think Dec. 21st would be a great birthday!! (one of my favorite days of the year...new season, yule, etc!! I agree with PineappleMama as well.
Both my family and DHs family have lots of birthdays around holidays...including a boatload of Dec b-days. For example, my b-day fall on 4th of July weekend (not a bad thing...I get a longer weekend for my bday) and DHs and niece both have T-giving bdays. My dad, two of my cousins, and DHs mom, DH nephew ALL have birthdays around Xmas/new years. Both of our families have the rule that b-days and holidays are celebrated separately. Even if it means that family from out of town is here for a holiday only (rare)...the cake/presents/birthday song is several hours separate from the holiday dinner, etc. If possible, they need to be a different day. Everyone has seemed happy with it.