That operation is called fmap, and it's characteristic of a functor. In Haskell, it's called liftM for monads, but they're really the same thing (and I think recent proposals are going to iron out this redundancy). fmap is one of the properties of a functor. All monads are functors, so all monads provide an operation with that signature.
But seriously, I guess I think it's not a Monad. The Monad signature (in this case) is
and I think what we really want (and what I said we wanted to do) is I don't know whether that's Applicative or Functor, but I don't think it's Monad.