@novi_homines: I think it's people who know Batmans' identity, work with him, were trained by him and sometimes adopted/brought up by him. Right now I think the Bat-Family is just the characters that were named in the poll.
Batwoman isn't one of the family because she was never trained by Batman, doesn't know his real identity and doesn't work with him. She isn't on Batmans speed dial for contacting allies, but all of the other family members are, and wasn't called to help during Night Of The Owls or Death Of The Family. Plus Batman and Batwoman don't seem to trust each other, and she recently said "I'm not a member of his cult.". Neither of them consider her to be one of the Bat-Family, so that's why she isn't.
I never liked or cared for Bats for Brains, but I did like Robin a great deal. I wasn't a big fan of his Nightwing persona so I grew to love Tim more since I was unaware of Jason until the animated movie came out. I wanted to vote for Jason but I decided to vote for Tim assuming he would be needing the votes. I love them both. I haven't decided which one I prefer at the end.
Of the ones listed, Jason (Followed shortly by Dick).
Though my favorites aren't listed (In order: Stephanie Brown, Terry McGinnis [does he count?], Huntress [Bertinelli] [does she count?], Cassandra Cain, and others I'm forgetting). I assume because they have yet to appear in the New 52. I didn't see the "Other" option.