Lily is cute and pretty.
Elle is good too, but I think I prefer
Lily just because it's a flower and I like naturey names/nicknames.
Some others:
Liz/Lizzy/Lizzie: I think this one is kind of boring and not the best, but it's common.
Ella,
Ellie, etc: Similar to
Elle Izzy/Izzie:
Can be good, but depends on the person.
Eli: I think this one is good for tomboys named
Elizabeth since it's a masculine full name. It's also cute in general IMO.
Elsa,
Elsie: Pretty/cute. I think these are good nicknames. Softer than
Elle.
Eliza,
Liza: Good and a little old-fashioned.
Lisa: I don't like this one as much but it's another one that could be a full name.
Libby: Cute but not the best IMO.
Beth: Sounds very friendly but not young. (Not "too old", just not young)
Lizbeth: Not everyone could pull this off, plus it's long for a nickname
Iza: Shorter and cuter