I don't like Benjamin especially, but the meaning of his being son of the right hand because it's a patronymic. I guess the girl version is Batjamin, daughter of Jamin?
I don't like the name as a whole, but I think a lot of cute nicknames could come from it. Ben, Bennie, Benji, Jamie, Minnie or Mina would be cute on a girl. It would also make a great middle name.