Working in "quantum mathematics" myself, I should tend to defend this teminology a bit ;) The term is clearly motivated by the usage in physics and, nowadays, is typically used in situations where you have a "classical" mathematical object (ring, algebra, group, whatever) which traditionally is viewed in a commutative context. Then the "quantum" version means to transfer things into a noncommutative context and see what happens.

Of course, this is all very vague, but why do you call groups "groups" and fields "fields"?
I guess, it is the intuition which makes this notion useful for the community. The intuition from physics is the transition from commutative to noncommutative, and I think that is really what people usually think if they hear from some "quantum blablabla" in math. So I guess, it is not a completely irritating notion :)

Quantumis catchy, historically motivated, sufficiently distinct from other terms that one usually knows what it is intended to mean when one sees is, considerably much better that "basic", and so on.... – Mariano Suárez-Alvarez♦ Mar 25 '11 at 18:17