Hi Tiffani,
One of my favorite memorable and kid-friendly restaurants at Walt Disney World is the Whispering Canyon Cafe at the Wilderness Lodge. The WCC has wonderful cowboy-style food such as ribs, chicken and beans. However, I'm not sure I'd characterize the WCC as "mellow." A big part of the fun is the over-the-top servers who bring a special down-home version of the Disney magic to your meal. You can read more about this
HERE.
Some other choices like this are the 50s Prime Time Cafe and the Sci-Fi
Dine-In, both at Disney's Hollywood Studios. The Prime Time is similar
to Whispering Canyon in that the servers play a big part in the meal.
They act like your family, telling Dad to take his elbows off the table
and asking Mom if she washed her hands after using the restroom. Kids
love it. At the Sci-Fi, your table takes the shape of an old-time car.
During the meal, you watch campy clips from old movies.
At Epcot, you might try the Coral Reef. The meal takes place right
beside the giant aquarium that is part of the Living Seas pavilion, you
get to watch the fish and divers in the tank as you eat. Other good choices at Epcot include Teppan Edo in Japan, where the chefs cook your meal right on your table, Restaurant Marrakesh, where your kids can learn to belly dance, and the Biergarten which features a German polka band.
In the resorts, Boma is fun. It is a buffet, which is good for picky
kids. But there are also some unique African dishes on the line that
are fun to try too. After you meal, you can walk around the resort and
see the animals on the grounds.
Hope this gives you some ideas for restaurants to try. Have a great trip!