I'm not sure what is kid-friendly....but during a recent weekend in Madison, we ate at
Ella's Deli - circus type atmosphere for the kids. Their kosher style deli foods are legendary, but it didn't do much for me.
Essen Haus - German (restaurant) Pub, fun atmosphere w/live entertainment at night, staff dresses in traditional plaids, etc, we went late at night for their wide selection of German beer, so I don't know what the food is like
http://www.essen-haus.com/
Dotty's Dumpling Dowry yummy burgers and such
http://www.dottydumplingsdowry.com/
Lazy Jane's- good for breakfast in an old house on Williamson, order in line, pick up your order at the end of the line. Weary Traveler on Williamson was also highly recommended.
Restaurant Magnus - probably too formal for kids, but very delicious experience
http://www.restaurantmagnus.com/