My 4 year olds play and enjoy the following Gameboy Advance games:
Spiderman Mysterios Menace
Spiderman: The Movie
Tony Hawk Pro Skater 2
Dave Mirra BMX
Many of the platform games require great timing and coordination to avoid instant death, which is tough for little kids. But the Spiderman games are not like that. The Spiderman games are quite easy to get the hang of, and even the youngest kids can play for quite some time before their health runs out.
The Skateboard and BMX games (and probably the snowboard one too) can be put in freestlye mode, so the kids can just zoom around pushing buttons to try and do tricks, without any pressure to score points or anything. The hard tricks are complicated, but the kids don't really care as long as they are doing some tricks.
Your 6 year old will do better than my 4 year olds, so you might also look at something a little harder to play like Mario Kart Super Circuit which is a very fun racing game. Also, my kids really love Megaman Battle Network but it is too advanced for them. They try, but alas you really need to be able to read for that one.
After a little practice your kid will be using the controller like he was born with it.
My kids have easily learned the Gameboy Color and the Microsoft Sidewinder Gamepad without any problems.
That's all I can think of right now. Hope it helps.