Well, it's worked for all the other women named Eliza who were once babies. I think it's more common to think of nickname on babies. Whether that's a shortened form of their formal name or just an affectionate name. DS is Theodore but we call him Teddy. In the first year of his life I called him Chunk-a-monk probably about 30% of the time too.
Our name if this one is a boy is William, nn Will. But sometimes it's hard to think of a baby being called Will. If that doesn't fit I'm not opposed to Billy or even Willie on a baby (I've known a few Willies- penis jokes aside).