So at chargen you'd have:
(akin to class. Personality determines the category of hat you could wear, style is your preferred hat type and the one that gives you the most bonuses. Perhaps you would just start out with a personality and choosing a style would come later on, like a specialization or PrC of some sort.)
Basic stats (point buy). Str, con, dex, int, wis, cha, if we're going by DnD, but it's pretty standard. Though I think we might want to add some more or split some up. Having 7 or 8 different stats means we can make hats a little more unique, stat wise.
Skills: General skills and "hat specific" skills. Hat specific skills would only be able to be used with a hat of the appropriate type.
Rolls: Xd4+Y, where X is the relevant stat and Y is the skill modifier.
Hats add to certain stats and give access to hat specific skills, as well as enhancing some general ones. They also add "hat tricks" which are special maneuvers, spells, or what-have-you, you can get more of these (and higher bonuses) once you specialize in a given hat.