there are some very-difficult-to-obtain and rare badges that, it seems to me, should come with some kind of a boost to base stats. For example what if "Team MVP" badge gave stun immunity, or +5% base power pip chance or +2% pierce. Stat boosts on the order of the kind of boosts that you sometimes find on a mount.
I really like this idea. The possibilities are endless.
I've also half-wanted a badge that you could get only if you've beaten one of every type of mob in the game (minus special quests for your school, of course). Something like that (because of how hard it would be, and because you'd have to redo bosses that are potentially really hard) should or would, under this idea, have a really special stat boost, IMO.
That would be cool if rare badges gave stat boosts. I don't see why it should only be the rare and hard to find ones. It could be certain badges in general.