Cute for a baby or old lady but not sure it suits a teenager, so I would agree with the other posters on Elizabeth, lots of choices for other nicknames. Reminds me of David Copperfield (Dickens, not creosoted magician)
I agree with the others - it's lovely as a shortening of Elizabeth, but not on its own. My little niece is Betty, which suits her at the moment as she is cheeky and curly, but when she gets older she can choose to be called a more grown up Liza or Beth.
I actually quite like it as a baby name, but would never ever call my child it as my guinea pig when I was little was called Betsy, after the rabbit my dad had when he was little! He called it Betsy because he got it in the year of the Queen's coronation.
I would also go with Elizabeth on the birth certificate.