I'm sorry. But, if this trip is for a sons 13th birthday. I think he'll find some of these suggestions a little boring...
I have a 13 year old daughter, and live in Mount Pleasant ( I work in Downtown Charleston, we live about 20 minutes from the city )
First of all, I hope you have good weather. We do have a few nice beaches at Sullivans Island, Isle of Palms and Folly Beach, but in February, don't plan on it being warm enough to enjoy them.
You son would probably like the slave market. Just because kids like to buy stuff. IMO, it's a bit like a straw market in the Caribbean, with lots of knick knack souvineers...but the basket makers are interesting.
Boone Hall is really nice. My DH grew up with the owner and we go over to visit, so I've never done it the "touristy" way, but the scenery is beautiful, there are horses, old slave houses and the Cotton Dock is lovely.
We also have the SC Aquarium downtown. If it's cold, I'd guess your son would enjoy that.
I do agree with Kaminskys. Best desert place ever. That and Marble Slab ice cream on the Isle of Palms connector.
For a bit of fun, try dinner at Tommy Condon's on Church Street. They have a live Irish Band playing after 8 PM ( they're off Mon and Tues ) Often a good laugh, and good music. Just don't stay too late on Fri or Sat night as they will lose their G rating a some point.
If you have any specific questions just ask.
ETA, I do agree with the Ghost Tour. One of the "touristy" things I really enjoy doing when we get guests in town.