As someone who has grown up surrounded by Daniels who go by
Benno and Ralphs who go by
Max, I'm quite tolerant when it comes to nicknames ;-)
So, yes, I think Effy for
Elizabeth is fine.
(I've recently read somewhere that Tetti is an old nickname for
Elizabeth, too. So why not Effy? After all, it's just a nickname.)
I know an
Evgenia who goes by
Effie.