So thinking about it, my favorite psionic character is actually fairly weak- Gil the Arm, from Larry Niven's Known Space stories. Gil was an asteroid miner who lost his arm in an accident. But he developed phantom limb syndrome, where he was sure his arm was still there- and then, to his surprise, found he has a telekinetic limb, with the same reach and fine motor control as his old arm and hand, but very weak. After much practice, he can eventually lift a pen.
He returned to Earth and got a cloned arm, but it didn't affect his wild talent, and he eventually went to work for ARM (a sort of future FBI/Homeland Security organization). For having such a weak ability, he gets a lot done with it.
I really like the idea of psionic wild talents, though they were anything but balanced in 1e/2e, and they were really weak to the point of not being worth discussing in 4e's Dark Sun. I know there are Feats now that can grant psychic abilities, but it might be interesting to expand on this notion.
I don't like that they are Feats though. You get so few of them, and I get the feeling a lot of players wouldn't take them if they weren't free (which at least one probably would be in a Dark Sun campaign setting). But at the same time, there's not a heck of a lot you can give up in 5e for an alternate ability.
I wonder how trading Hit Dice for power would work? I'm sure my play experience is unique, but I've rarely had to spend Hit Dice to regain hit points, so I think that might be a worthwhile tradeoff, but I'm not sure.