Tastes vary so much. I mean, some people will just go for the sports genre. Other than that, with the consoles you have I could make a list about a mile long of things I enjoyed at that age (as a young Christian teenage guy

).
Wii: Super Mario Galaxy!!!!!!!!!! I can't imagine any red-blooded human being not enjoying that, and it's totally family friendly.
(Wii plays all GameCube discs if you have a GC controller and memory card): Metroid Prime (!), Super Smash Bros. Melee (everyone loves this one).
The Wii also has a "virtual console" (you'll need a "classic controller"--$20) and if they are agreeable to older games, they go for $5-$10 apiece for download--I enjoyed some of these when I was younger and they are much easier to "pick up and play" for short periods than newer games. In my mind it's an awesome value: Pretty much anything out of the Mario or Zelda franchise are just about impossible not to like.
http://vc.nintendolife.com/gamesAnything rated 8 or higher is a safe bet, and I can hardly think of more than a handful that you could possibly call "sinful" (R-rated violence, swearing, "suggestive themes" etc). Some of my favs (from when I was that age, too): Super Mario RPG, Super Metroid, Mega Man 1, 2, 3 . . .
As for the PS2 also plays PS1 titles which had many of my favorites: I was into everything "Final Fantasy," "Ico," "Kingdom Hearts,"
Xbox360 is good for old Xbox games, in my mind. Star Wars: Knights of the Old Republic, Burnout 3,
DS: Chrono Trigger, Mario Kart DS, Legend of Zelda: Phantom Hourglass (GBA slot) Final Fantasy VI, Legend of Zelda: Minish Cap, Warioware.
The great news with games is that it's easy to get bang for your buck these days: the old stuff is great, the newer games mostly aren't, and "old"=cheap=more than anything more than a year old.

I could go on but those are some off the top of my head.