I really like
Betsy (and Betsey as it's spelled on my family tree) and this that it's sweet. It is a little old-fashioned in sound but using it on a little girl would bring new life to it and freshness. I think it's cute and spunky and sweet.
I know a lady named
Betsy who has is thin, blond, and pretty and the name always makes me think of her or a little girl with curls. I think it has a very young sound to it.
But I don't think I'd use
Betsy as a name by itself but as a nickname for
Elizabeth incase later in the girl's life she looks or feels more like a
Liz,
Lizzie,
Eliza,
Beth,
Lisa, or
Ellie. :)