DH thinks maybe we should go with Elizabeth if it's going to be Eliza to give her more options
The thing is, the only extra "options" she gets are Beth and Elizabeth. Eliza can already be shortened to Ellie, El, Liz, etc.
And calling her Elizabeth on the birth certificate just to use Eliza in day-to-day life seems pointless, since Eliza is actually a real name of its own.
I'm a big fan of Eliza but not so much of Elizabeth.