1. Psionic capabilities are innate to humanity, but guided by extremely recessive (if that makes sense) genes that are quite rare. But, if you get Joe the Mildly Esper to marry Jane the Unusually Lucky, and continue the process, you might after a few generations get Bob the Telekinetic.
I just read 'To the Magic Born' by Christopher Stasheff, and of course, Stephen King wrote 'Firestarter'.
2. M.C. Escherian techniques. I've used this in some of my games. Look at a picture of something physically impossible, and hold the whole thing steady in your mind, and you will activate unused parts of your mind.
3. Pschyosurgery (accidental or otherwise) to activate unused parts of the brain. This can be invisible powers of a telepath, or it can by the crude gadgetry of drills and electric jolts.
4. Mental discipline. Meditation. Skills such as taught by certain monks.
5. Meditation based on the understanding of the universe. The brain is a radio that receives the signal, and sends back to the spirit realm...this is what a human is. But, such a capability requires the ability to affect quantum states at the micro level by thought. It is possible to teach the brain how to affect mass areas and achieve far larger effects than moving electrons.
6. Focused imagination.
7. Degradation. I did this in one world where acts of moral decay freed practitioners from their mental bonds, and allowed them to access their own capabilities.
8. Pain. Sufficient and vast pain, often emotional, can awake psi powers.
9.
Of course, the theory in connection with player characters seems to be one of two notions, both of which start with the assumption that anyone can do these things if they figure out how, just like (nearly) anyone can learn to walk a tightrope or solve a problem in calculus. For some player characters, they've never done so presumably because those things are not possible (or at least are difficult beyond the ability of the ordinary person to accomplish) in his universe of origin, but now that he's left that he can tap into these abilities. For others, well, they claim they already have those powers, and who am I to argue? So they do, and there's nothing to explain.
--M. J. Young