How are
Ellie and
Ella and
Elle not related to
Elizabeth?
Elizabeth starts with
El, it's right there.
Ella's pretty nicknamey, it has no separate etymology other than a nickname. I know people use it as a full name now but... it comes from names like
Elizabeth and
Eleanor...
Lily isn't so much of a stretch. e-LI-zabeth, and then you get kind of stuck on the
Li- sound.