My little cousin is called
Libby but that is her full name rather than it being a nickname as it is usually used. I dont see any problem with it being used as a name by its self. Ive only ever heard of it eing short for
Elizabeth or
Elisabeth but i guess
Olivia could work(Although i know a lot of people called
Olivia nn
Livvy).
Isabella and
Isabelle in their various forms could also strech to having the nickname
Libby i guess!!