I grew up with Mt. Rushmore in my back yard. If you want a central location to all of the Black Hills, Rapid City is the best location. Rapid City has around 75,000 people. If you want a quieter time I recommend Edelweiss Mountain Lodging. They have Cabins in the Black Hills in between Mt. Rushmore and Lead/Deadwood.
If you are the camping type, stay a few nights at Rushmore Shadows campground. They sell a campground membership (actually how I got my start in the timeshare industry). If you sit through a presentation the nightly rates are a bargain. Tell them John says hello.
If you want to stay in Rapid City, the Hotel Alex Johnson is the oldest hotel in the city, and it is right in the middle of downtown. There are a few hotels with water slides on the east end of town.
Make sure to check out the Badlands National Park about 80 miles east of Mt. Rushmore.
Feel free to PM me if you have any specific questions or off the beaten path items to do.