The beach is the cheapest thing, that's where I'd want to be if i were a kid.
Buy a frisbee, a shovel and sand pail, a cheap boogie board and a kite and set them free.

There's a children's museum that's not expensive and kids could probably blow a day in there (mom and dad may get bored after a while)
- interactive exhibits, giant legos, a 'ship' made from a waterbed so you slosh around in it, arts and crafts area, science experiments, they change the exhibits and I haven't been in a while.
There's small parks scattered around. If you like to fish, bring a pole and you can fish off of most of the piers. There's a nominal fee, it used to be less than a dollar but been a while since I've done this. The Surfside Beach and Garden City piers both have little tackle shops where you can buy bait. (you can also buy and maybe even rent poles/tackle)
Beyond that Putt-putt (pretty cheap), the amusement and water parks (not horribly expensive), on up to the shows and special venues - the aquarium, IMAX, Ripleys museum, Magic Quest, Alligator Adventure, Medieval Times, Dixie Stampede...there's a lot to do. Most of these places have websites and most offer coupons. There are usually fireworks at Broadway at the Beach at least one night a week during the summer, free to watch.
Depending on how able you are to set a budget and say "no" to their pleas for more quarters you might be able to do an arcade trip fairly reasonably.