Psi Lab (EU2012)
The Psionic Lab will allow you to test your soldiers for psionic powers. You may test up till 3 soldiers at a time, upon which they will be unable to join a mission during 10 day testing time.
The chances for getting a psionically gifted soldier seems to be random, but with a higher chance the better willpower the soldier has. Because of this, the current assumption is that a Colonel that started out with 52 psionic powers and ranked up while the [Iron Will] was active (with the maximum 12 Will increase per new rank) has the best chance of getting the gift.
A soldier with 50 willpower has less than 1% chance of being gifted. 
Using Save and Load, it was found that the chance of being gifted is calculated immediately when the soldier is put in the Psi Lab. This means each time the same soldier is removed and re-added to the training tab, he/she would have a recalculated chance of being gifted. This is best done when you have ten days of uninterrupted peace.