Miami with kids
What do your kids eat and what part of town will you be in? Our kids are a couple years older than yours, and until about a year ago eating out basically meant finding chicken fingers, pizza, or plain pasta. They've become more adventurous of late, but I know what it's like to hunt down something they will eat at someplace you'll satisfy chowish desires too.
Without knowing quite what you're looking for, I can still make a few suggestions, which are mostly on South Beach:
Big Pink - big menu of comfort food-type stuff, located down toward the south end of South Beach . Breakfast is especially good (and served any time), but there's lots of decent+ food throughout the menu. My wife likes the Southwestern Chicken Salad, their burgers are good, I like the rotisserie chicken salad sandwich, the yellowfin tuna sandwich, the BBQ chicken wrap. Kids menu too. Some outside seating. Service can be slow.
Cafeteria - similar kind of place to Big Pink, on Lincoln Road. Maybe a bit more upscale than Big Pink, which is styled a little more like a diner. Again, breakfast is very good, the rest of the menu can sometimes be a bit hit or miss, but lots to choose from to make folks happy. I've liked their calamari salad, the soup & sammie combination w/ tomato soup and grilled cheese w/ bacon & tomato (talk about a nostalgic dish!), fish & chips were good. Also has some outside seating on Lincoln Road. Also can have somewhat lackadaisical service.
Mr. Chu's - if your kids eat Chinese, this would be a good place, especially for a dim sum lunch. We started taking our kids to dim sum around the ages yours are, and they now beg me to go every weekend. I think the smaller portions may make them a little more bold, and getting to peek into all the steamer baskets in the push carts is fun (plus, I always let them get the sweet stuff like the custard buns too). They now regularly get roast pork buns, soup dumplings, and will try at least a couple other things every time we go. The dim sum at Mr. Chu's is excellent. Their regular Chinese menu is also quite good, but we usually go for the dim sum. On Washington Ave. & 9th St. on the beach.
PIzza Rustica - quite good pizza by the slice, with a few locations on South Beach (two on Washington Ave., one on Lincoln Road). I usually like my pizza thin and crispy and this is more of a thicker focaccia style crust, but it's still delicious. Lots of different topping choices.
Rosinella - good, simple Italian food. Toward the east end of Lincoln Road with lots of outdoor seating. The menu from the website below looks short because they usually have a full page of daily specials. See if they have gnocchi, they're excellent.
Garcia's - responding to your "uniquely Florida" request, this is a little seafood market & restaurant on the Miami River where they have the best fish sandwiches in town (usually grouper or dolphin). It's a bit of a dumpy place, but you can sit outside in the back and look out on the river - which is not nearly as scenic as the ocean, but it's a working river w/ commercial boats out there, and I think the kids will get a kick out of it (I still do). Located at 398 NW North River Drive.
Scotty's Landing - OK, the food is no great shakes, but it's a great place to hang out and look out on Biscayne Bay. Conch fritters, peel & eat shrimp, fish sandwiches are all decent.
Cuban food is often a good bet for the kids and is also clearly a Miami thing. Chicken breast grilled or empanizado (crispy breaded) with black beans & rice often will work for the kids. I usually go to Sergio's (Coral Way & 32nd Ave.); Versailles (8th St. just east of 37th Ave.) is a local institution (and an over-the-top place with all the mirrors and chandeliers). I don't eat Cuban that often and others may have better suggestions.
Another taste of local flavor would be an Argentine parrillada. Our usual, just because it's close to home, is Las Vacas Gordas, on 71st Street (Normandy Drive) on the beach. An overabundance of grilled meats, plus you can get pastas for the kids if they're not voracious carnivores yet. After you can go across the street to Dolce Vita, where they have excellent gelato.
If you're looking for choices in a particular neighborhood and/or particular types of food, can attempt to hone these suggestions a little more.