Darn, I was going to make a very similar post but never got around to it.
An idea I had when the N64 came out with its multitude of colours of controllers was that I was thinking there should be some way for the controller to tell the game what colour it is. Then when playing multiplayer games, by default, the characters could be clad in the colour of the controller (by default, of course, not limited to that). With internal memory, you could do one better - and have the controller remember your personal favourite colour (you would set it by RGB values, and then you could have one of 16.7 million possibilities with using up only 6 bytes).
Depending on how the sensitivity works, you could have different settings for different genres. A multi-genre game might have a code in the game saying that you're entering a racing stage, and then a shooting stage, etc., if you require different settings for each one. If too complex (though not really) I'm sure there could be simple settings like whether or not you want the Y-axis inverted by default, etc.
Maybe a quote/comment/personal greeting (like the DS has).
6 kb (or is it 6kB? Either way it) isn't a lot to work with, but depending on what online features are available, maybe your mini-profile could also show your favourite game(s), the game(s) you want people to challenge you to, or a score or percentage completion or accomplishment in a certain game that you want to show off. For example, you may not believe me if I told you I made it to 999,999,999 Bells in Animal Crossing without using an external cheat device (which I did, BTW) but that would sure be an accomplishment I'd show off on my profile. Of course, you'd have to be sure external cheat devices could somehow be detected by the system...*sigh*
There could maybe also be a personal code or some small snippet of personal information that might be used in a game somehow, either for laughs or as an actual key to a puzzle. I know Mario Party Advance asks you some questions (like "what's you secret?" - creepy) but I don't know what it is for. Or the system could ask you some moral questions (like in the Ultima games...do you value honesty or compassion more, ie. you're delivering an uncounted bag of money when you come across a beggar. Do you give him some money (in compassion) knowing it won't be missed, or do you continue on your way with the full sum of money (because of upholding honesty)?) and maybe this will determine how good or evil your character will be prone to become in any game without having to do the same process in any game. (Though maybe not, that's part of the fun.)
One thing that ISN'T part of the fun is recreating my face in a gazillion different games. Perhaps it's too much to ask that all Tony Hawk games read the same "face/character creation" file and that EA games do the same, but one per company would be nice. Then again, with only 6 kB perhaps that's better left to the internal flash memory or the SD cards. Some kind of singular face/character thing. That would save a lot of time.
Just some ideas of mine.