What do you think of the name Bitty?

I met someone with the name Bitty. Her legal name is Elizabeth, but everyone calls her Bitty. She's a teenager, by the way. Is this a common name? What do you think of it? It's starting to grow on me, but the girl won't be a little girl for much longer. Do you think it's just a little girl name, and Bitty should go by Elizabeth or Liz or Beth as she gets older?