You have a few options.
First, you may want to contact the provost marshal at Ft Jackson. They may allow you to register the firearm then store it at the armory on base. Ft Jackson does have a range (rod & gun/skeet club) so they clearly have some provision for allowing privately owned weapons onto the base.
You can contact Shaw AFB Security Forces. Shaw is about an hour down the road. They also have skeet club as well as a pistol/rifle range and thus must have some provision for allowing privately owned weapons past the gate. They may allow you to check the weapon into their armory for the duration of your stay in SC.
McEntire National Guard Base is also right down the road -- they may have an armory you can stash the firearm in.
You could stay in a hotel off base and leave the firearm in the safe in the room. This is probably the least secure option. It is an irony that gun control efforts increase the risk of theft. I'd make sure that the firearm is insured.
You could secure the firearm in your car (locked case in the trunk), park the car off base, and catch a ride to your son's events on the base (or walk from the gate). Same caveats as the hotel regarding theft.
You could rent a "self store" unit in the local area near the base. There are some really cheap units. Then you just stash the handgun in the self-store unit. This costs money, but can probably be done for less than $50.
Good luck!
