Custer is a good place to base out of for the south hills. It is close to the big time attractions (Rushmore and Crazy Horse) and also lesser known, but excellent ones like the Custer wildlife loop, the mammoth site near Hot Springs, Wind Cave, Jewel Cave, etc.
You can't go wrong with any of the Custer State Park campgrounds, but I agree with Booster that Stockade Lake South is probably the best if you are looking for one off the beaten path. If they are full, Bismark Lake campground (a forest service facility) is right across the street and is a little cheaper and nice, but without much for hookups. I've stayed at Big Pine which is on the west side of the City of Custer and is very convenient. As mentioned above, it is a nice campground. I've stayed at many other campgrounds around Custer and have never run into any that I would stay away from.