Note: Everything I'm discussing here is located downtown - because that's where the touristing is
Not sure what your price range is, but there are quite a few options. Most of what you'll find in downtown Charleston (and the whole Charleston area for that matter) is Low Country Cuisine. Is this something that your family will enjoy? Most folks like it, but is it something that they can get where they live? If not, than I'd definitely recommend giving it a shot. They'll enjoy a walk on King's Street and Market Street; everyone loves that area (besides Benke). SNOB, High Cotton, Oak, and Magnolia are good restaurants but a little on the pricey/high end side (and possibly less than kid friendly). Poogan's Porch, 82 Queen, and Husk are all right next to each other, stationed between King and Market, and all represent Low Country Fare pretty well with a reasonable price (for the area). If Low Country fare is not popular with the crowd, give Poogan's smokehouse a look; for the area the price is reasonable and they've been good at accommodating larger parties. And also on that note, Lewis Barbecue, Home Team BBQ, and Rodney Scott BBQ are in the "meat packing" district and are all excellent (though Rodney is from Upstate, so you might've had his whole hog stuff). Lewis BBQ has the best brisket I've had on the East Coast and Home Team smoked wings and white sauce are unbeatable (I'd leave the rest of their stuff though, there is better).
Market Street has a bunch of handmade oddities that scream South Carolina. Try to steer clear of the sweetgrass baskets, though, they are overpriced and you can find it cheaper elsewhere. Snack on some pralines from River Street Sweets (just listen for the woman screaming "Free samples!"). I'd enthusiastically recommend "The Old Dungeon" for lots of Charleston and Civil War history. Also, there are several OG BMFIC houses in the city that were restored to their pre-Civil War era prominence that you can tour. They were pretty fascinating to us, but the one that kinda blew us away was The Heyward-Washington house. Nathanial Russell House was also pretty cool. The Charleston Museum is a good option as well as the Charleston Aquarium (make sure to see the Turtle Hospital). And lastly, I'd suggest hitting the Hunley Submarine. The Hunley museum is located near "The Codfather". Having just gotten back from England, I can honestly say that The Codfather has better fish and chips than you will find ANYWHERE (I'm a fat guy that loves food, don't test me, you will lose). Angel Oak is cool, but it's a hike to get to - it's way out on Johns Island. If you head out that way, though, because it IS cool, I'd recommend "The Southern General" for good lunch food.