I like Elisabeth Kaye! It's classic and will age well. My little sister's name is Elizabeth, but we call her Lizzie.
Now, nicknames...
Ellie
Elle
Eli
Sally (stretching it a little)
Betta
Lily (stretching it a little)
Elsie
Elsa
Sabe (SAH-bay, like, with an accent over the 'e')
Bethie
Lissa
Lissy
Lisa
Do you like any?