Charleston County runs a family-friendly, fairly boisterous campground at James Island County Park (871 Riverland Dr., 843/795-7275, www.ccprc.com [1], $31 tent site, $37 pull-through site). A neat feature here is the $5 per person round-trip shuttle to the Visitors Center downtown, Folly Beach Pier, and Folly Beach County Park. The Park also has 10 furnished cottages for rental, sleeping up to eight people (843/795-4386, $138 a day). Reservations are recommended.
For more commercial camping in Mount Pleasant, try the KOA of Mt. Pleasant (3157 Hwy. 17 N., 843/849-5177, www.koa.com [2], $30+ tent sites, $50+ pull-through sites).
Links:
[1] http://www.ccprc.com
[2] http://www.koa.com