Unlike the first example, this character actually has an above average attribute (Charisma) and only one below average attribute (Strength). The highest class-related attribute is Wisdom, so I'll make her a cleric. Of Chaos. Meet Kyraptica, Priestess of Phlaagra.
Rolling All Dice, my results were d4: 3; d6: 5; d8: 5; d10: 6; d12: 1; d20: 1.
The d4: Psychic Fuel
With a roll of 3, Kyraptica's Psychic Fuel is her Constitution, which is average. She is fortunate she does not have to draw upon her Strength to fuel her Power.
The d6: Psychic Focus
With a roll of 5, her Psychic Focus is Intelligence and Charisma. Her Intelligence is average, but her Charisma has a bonus of +1, which is added to her roll whenever she uses her Psychic Power.
The d8: Psychic Rejuvenation
With a roll of 5, Kyraptica restores her lost Constitution by merely clapping, for which she regains 1d4 points. She may do this once per day. (She can clap all day if she likes, but she will still only regain 1d4 points.)
The d10: Psychic Enhancer
With a roll of 6, her Psychic Enhancer is a gong. Kyraptica's cult makes use of the gong in ceremonies, so it's a multipurpose musical instrument. The reverberations caused by striking a gong intensify her concentration and give her a +1 bonus to Psychic Power rolls.
The d12: Psychic Failure Consequence
With a roll of 1, any failed Psychic Power roll results in psychic paralysis for 3d6 rounds, during which time she is effectively non-psychic. This may seem like a minor setback, but in the presence of another psychic it is tantamount to revealing that you are dangerous and then dropping your weapons, shield, and armor.
The d20: Psychic Power
With a roll of 17, Kyraptika's Psychic Power is Pyrokinesis, which suits her fiery temperament. This is the paranormal ability to produce and control fire with the mind. Control can be maintained as long as complete concentration is possible. Anything that interferes with concentration requires another Psychic Power roll to be made.
Kyraptica, Priestess of Phlaagra, has the gift of Pyrokinesis, which marks her as chosen by her deity to spread the message (and Chaos, which is pretty much the same thing). Her use of this gift drains her health, but she can revitalize herself by merely clapping her hands. Her control of the gift is guided by the strength of her personality and by the sounding of a sacred gong, which she carries with her. Should she stumble in the use of her Power, however, she loses that Power and all of its defenses until she can summon her resolve.
[Edit: I changed the term "Wild Talent" to "Psychic Power." 2015-05-13]