I don't think it works. When I think Eliza I think - Lizzie or Izzy, maybe Ellie. You can use whatever name you like though! Everyone guesses my name incorrectly based on my nickname.
I'd use it. As a pp said, you can use any nickname you like. But at least with Lil the 'Li' part of it has been taken. People will probably say "is that short for Lilly" a lot but taht's not really much bother. I mostly go by Ellie but I get asked "is that short for Eleanor" a lot, despite Ellie mostly being short for Eleanor- people like to check. It isn't a big deal to say "yes" and it won't be a big deal for your LO to say "no, Eliza".
Lily is my all time favourite name!