They call this "character dining," and they have whole restaurants dedicated to it in each park. There's Cinderella's Royal Table and the Crystal Palace in the Magic Kingdom, Tusker House in the Animal Kingdom, Akershus and Garden Grill in Epcot, and too many places to list within the resort hotels. Pick your kid's favorite characters, and you'll find them in the restaurants within that park.
You're not alone in wanting this, it's actually a pretty common question. Unfortunately, that means that the answer is always changing, as people shift from area to area. Right now, the best answer seems to be the West Oaks Mall. It's out of the way and far enough from the main attractions in Orlando to discourage tourists, but then again once word gets out, you'll be looking for a new place all over again.
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