This is one of the great questions for the ages. If you believe in evolution, it follows that "Eve" had at least some semblance of a belly button - probably similar to that of a chimp or other ape. If you believe in creation it could go either way, though I believe God would have imbued Adam and Eve with belly buttons so as not to creep out Cain and Abel.
I don't believe Adam and Eve had belly buttons, but it's not because I believe that a God made them that way. I don't believe that they existed in real life and are part of an ancient myth, just like Zeus and Hera, so it would be awfully hard for either of them to have belly buttons.