These are games that my kids like to play. But... I got tired of my kids wandering with the mouse pointer to a link that takes them outside of the game. I changed the games to remove the links and to be full-screen, which also provides more of an immersive game-playing experience. The games are now more kid-friendly, at least for my kids. Links to the game sources are also provided.

Sketch's World (a game to help kids learn their multiplication tables.)
Added Apr-28-2011

Penguin Jump (a competitive online game to practice multiplication tables.)
Added Apr-29-2011

Crumbled 2
Edited Dec-31-2009

Fancy Pants Adventures
Added Feb-18-2009

N
Added Feb-18-2009

Mario Star Catcher
Added Mar-29-2008

Skatefall (like the old Pitfall classic)
Added Mar-29-2008

Elmo's Keyboard-O-Rama Added Jul-28-2007
New Super Mario Bros
For such a small machine, the DSi provides surprisingly good graphics and stereo sound. See the game in the video below...
Claymation - a flash claymation experiment Added Jul-08-2007
Dynamite - a flash cartoon experiment Added Jun-14-2007
Garry's Mod - flash video captured from the PC game
Elebits - flash video captured from the Wii
Rayman: Raving Rabbids - Wii game from France
Canabalt
This game reminds me of Knytt Stories (another good game) but with more adrenaline. The music that accompanies this game is good enough reason to download it. Perfect sound effects. My kids like the game. I think the game is good for any age level since all you do is jump. Literally, that's it... no running control, no having to change directions.
Tip: make full-screen by pressing Esc. To get out of full-screen mode, press Alt Tab
Download CanabaltFrets on Fire
This guitar-playing simulation is quite fun but has a learning curve. (Similar to Guitar Hero.) Once you understand which buttons on the keyboard to press, this is a beautiful game and the music included is good stuff. Has a music editing and importing tool. Might be some fun to add your own OGG music files and play along.
Download Frets on Fire