Lily is not usually a nickname. It stands alone and has for a very long time. Supposedly it started out being a nn for
Elizabeth, but I think that today using it that way is trying too hard.
I don't like
Lily very much at all. You're right that it's adorable, but it's adorable the way twin pigtails tied with pink bows, dresses with pinafores and lispy baby talk are all adorable. Meaning, it's sickly-cute and doesn't age well.
I always picture
Lily as somebody's adopted Chinese daughter (for some reason
Lily is very popular for adopted Asian kids) who owns thirty-four different princess costumes and throws a fit if she isn't allowed to wear them every single day.