I'm not sure having a Monoid class is actually useful for anything - but if we must have it, there seems to be little better possible name for something so vague.

IMO the Monoid class is useful since, if you define mempty and mappend, you get mconcat for free. I don't see what the problem is. Most people will accept Functor, as it used a lot. Monoid might be less used, but if you reject it, then by principle you must just as well reject Functor, and any other type classes.