I love Elizabeth and would probably choose it out of the two, but I do like Delphine too. It's unusual without being weird, has a sort of Continental chic about it, and always makes me think of delphiniums which are very pretty flowers.
Delphine Elizabeth would be a nice combination if you would go for it. She'd still have the option of using Elizabeth or one of its many short forms if she didn't feel like a Delphine.