The 1991 hit Lemmings was one of the best and most popular games of its day. Luckily, it's coming back for the Sony PSP and PS2. Control the cute Lemmings by giving them abilities, like climbing, digging and building, enabling them to find their way through a variety of levels. Other fun games with cause-and-effect problem solving themes are Chain Reaction, Eets and Armadillo Run. (At the time of this writing, the Lemmings site above by Sony is often slow and unresponsive. Wikipedia has another excellent Lemmings resource.)
From Mario to Zelda, Nintendo excels at imaginative, family-friendly video games. If you've got a handheld Nintendo DS, take advantage of non-violent games with multiplayer capabilities:
Solve puzzles and test your memory with the most important tool you have, your brain: Give your noggin a fun workout with Nintendo's Big Brain Academy and Brain Age. Or check out Animal Crossing, where players live, work and improve the town they live in. Design and wear your own clothes. Pick furniture and design artwork for your house. Visit the museum to view exhibits, learn about fossils you've found, donate, or go to the observatory to view and create constellations. Talk and write letters to friends. After you've customized your town, invite other players to visit. Or visit and explore their towns.
A Force More Powerful is a single-player, turn-based "game of non-violent strategy". Players learn how to defeat dictators, solve human rights violations, topple corrupt political regimes, all with non-violent methods. Modeled on real-world situations, AFMP is more than just a simulation. It's a way of educating players how non-violence is a powerful tool in fixing social, racial, religious and political problems.
"Dance games" like Dance Dance Revolution and In the Groove are found in some arcades and most home consoles and PCs. "Dance" by stepping on a special footpad in time to music. A great workout and a lot of fun, your toughest competitor will be yourself. A free version of DDR called "Stepmania" is available for the PC.
Civilization IV: Start with a small group of settlers. Pick a good spot, and build a city. Use the city's growth to further scientific research and exploration. Build more cities. Meet other civilizations and negotiate for peace or war. While conflict in Civilization is possible if desired, it's not required to win. Learn about and control scientific progress, political systems and foreign diplomacy, and play one of the best epic world-building games of all time.
Good luck!