What watching the olympics with your kids might teach them:
Teamwork and Sportsmanship.
This is a no-brainer. Watching a team sport is an awesome way to show kids how players work together -- and work hard -- for a common goal. As for individual sports, it's also an opportunity to talk about how solo sports are cool, too. Individual athletes are supporting their entire country's team and cheer each other on. (Don't forget how solo athletes work with their own team (hello, Dara Torres) to even get to the Olympics). You can also show your kids good (and bad) examples of how players treat other teammates and competitors. Lessons are always more memorable when you can see something in action.
Geography and Culture.
Keep an atlas handy while your watching the games. If you have younger kids, break out the map and show them where competing countries are located in relation to you. Talk time differences, weather, how you'd travel there and what you would do and eat (also something you can ask your older kids about...nothing like an impromptu geography quiz!).
Supporting your military at war is a common example used when folks describe the phrase "patriotism." But with kids, it's cool when you can show that patriotism doesn't equate with violence or fighting. Supporting your country's Olympic team helps teach kids to be proud of where they live. It's also great to teach your kids that it's okay (and fun) to support other countries. I remember cheering the teams from Ireland and El Salvador when I was younger because they represent my heritage. But I also loved Brazil (because of their kick-ass flag...little kids and their wacky rationale).
Variety of sports.
Your kid may love basketball or gymnastics. But what about watching something different, like water polo or soccer. Trampoline anyone?