What would happen if i went back in history and deleted to Eve???
Who?
More seriously, eve is NOT history.
Even if the Bible's Eve didn't exist, there was logically most likely still a living being on earth, at some point, that could be classified as the first female modern human.
Interpreting it as such, I think destroying her would have a more drastic affect than destroying the Mitochondrial Eve linked in OldTrees' post, as modern humans would not really have existed yet. I don't know relatively how much, though.