If you decide to stay outside the park, IMO the best choice is West Yellowstone. Lots of motels and only a mile at most from the west entrance. The majority of thermal features are between Madison Junction and Old Faithful. Madison is 14 miles from West Yellowstone and another 16 miles to Old Faithful. If you are short on time, you may want to limit your tour of Yellowstone to the lower of the two loops as the majority of points of interest are there.
FYI, we met our son and several friends 3-4 times for Yellowstone and Grand Teton. He flew into Jackson Hole airport and they rented a car to tour the area.