It's incredibly useful, since you can just go into Hoogle or some similar tool and query a function that:

    Monoid m => (a -> m) -> [a] -> m

1 - Well, the autodoc write those comments for you.

Associativity is a boring word, because high-schoolers have seen it.

Eh. Your patience here is admirable.

Every time `instance Monoid Foo` occurs in the Haddock documentation of a Haskell datatype which is pretty often!

> Does the frequency with which this occurs justify introducing new terminology?

"Justify" according to what set of criteria? Personally I prefer it.

> you artificially inflated the wording to make "x is associative and has a neutral element y" look more complicated than it actually is

So your suggestion is

* baz is associative and quux is a neutral element for it

Really, if you're going to go that far you may as well go all the way and just say

* Foo is a Monoid under baz and quux

> while not even naming in your "short" case what the operation and neutral element are.

Well, in the Haskell world they're implied by the typeclass instance, but I take your point.