I like the idea of psionics as an expression of inner power, manifesting your desires into reality by sheer, tenacious, razor-edged force of will.
I don't think it's out of place in a game like D&D for a character to have that ability, as distinct from arcane expressions, which channel the innate "magic-ness" of the world, and divine expressions, which are boons granted as rewards from the gods. That triumvirate seems pretty sound to me.
In my own campaign I happen to have categorised psionics as an infusion from the Far Realm, a place so different from everywhere else that it has its own sources of strangeness.
What is it about psionics that implies they *shouldn't* exist in fantasy? Was the original Gygaxian implementation steeped in meta-science or something? I don't really recall.