I would definitely not say that it's a common diminutive for
Elizabeth.
Ella is common on its own. I actually don't like
Ella,
Ellie,
Elle or
Lily as nicknames for
Elizabeth. I just don't see how you get
Lily out of
Elizabeth and
Ella,
Elle and
Ellie just aren't related to
Elizabeth in any way so I find it extremely odd to use them as nicknames. I wouldn't use
Ella as a nickname for
Eleanor either, now that I think about it. I'd just name her
Ella if I wanted to call her that. It's not like it's nickname-ish or anything. To me using
Ella as a nn for
Elizabeth is like using
Anna as a nickname for
Andrea or
Angela or something. To each their own :)