The longstanding myth was that he was given the name by a ouija board but he denied this saying that he picked the name to conjour up a picture of innocence.
I might be wrong here but my recollection is that Alice Cooper was originally the name of the band not the individual and then over the years Mr Furnier became (inevitably) known as Alice himself...I think!
The same thing happened in the 1960s with the group Manfred Mann which although using the first name of it's founder and leader (Manfred), Mann was not his surname although most people assumed it was at the time.
And I read somewhere that the same thing has happened more recently with Bob Sinclar, the front man now having assumed the name originally given to the band.