If you pick just one of the three stats, which will always provide bonus AP's, that stat becomes much more useful to all classes.
You therefore hand a significant advantage over to classes which already benefit from having a high stat in that attribute. Since you are focused on mental stats, casters will automatically have an advantage. If you select INT, then Wizards, Witches and Alchemists benefit. If you select WIS, this will be great for Clerics. CHA favours Oracles, Sorcerers and Paladins. Classes that don't benefit from the stat you choose will be disadvantages, either forced to devote stat resources to that characteristic, or stuck with less AP's since they don't put much in that stat.
So I would modify Billd91's suggestion to either "the lowest of the three mental stats for that specific character" or having all three feed into AP bonuses, given you are determined AP's must be determined by mental stats. Basing the bonus on the lowest of the three, or the average of the three, will avoid giving a huge advantage to classes which already benefit significantly from whichever one of the three you choose.
It's important to recognize that you are adding an incentive to whichever stat you choose. If, for example, INT becomes determinative of AP, expect that Sorcerers will be pretty rare - Wizards and Witches will make much better arcane casters, and Maguses probably become a more common "warrior" choice. Elves and Tieflings become good race choices, especially for rogues who value that DEX bonus anyway.
If you choose WIS, don't expect to see many Oracles - Clerics will have a significant advantage. Gunslingers become an excellent Martial choice, as they benefit from high WIS. Probably Rangers as well. Dwarves become a solid race choice, since they get bonus WIS and CON, two stats every character now needs. That will further demotivate CHA-based casters, since that's the Dwarven penalty.
Pick CHA and you'll see a lot more Oracles and Sorcerers, with Paladins becoming an excellent Martial choice. Dwarves become much less frequent, but expect more Gnomes and Halflings, especially as casters who don't need much in the way of STR.
I think you're approaching this the wrong way. It seems like you've already decided you want to pick one of INT, WIS or CHA (ie you aren't open to having all three be relevant, or using a "lower of" or "middle of" the three approach. That means whatever choice you make will favour some classes, races and builds, and provide a disincentive to others. With that in mind, I suggest the question is which stat you want to push the players toward, and which one(s) you want to push them away from. For example, if you want to see more interaction-focused characters, more small characters, less Dwarves and less wizards, witches and clerics, choose CHA. Assess which character builds are advantaged by, or disadvantaged by, each stat, and select the one that provides advantages to the kinds of characters you want to be more prevalent in your game, and the greatest drawbacks to the characters you'd like to see less of.
