Here's a recent thread regarding camping near Space Center Houston.
Please note that Space Center Houston (which is next to JSC) is different from Johnson Space Center. You cannot drive into, or 'tour' JSC except for a limited area called 'Rocket Park'. There's a constant stream of RV's being turned around at the JSC gates. That said, Space Center Houston is a great place to visit and they have a tram ride that takes you to JSC and Rocket Park.
You may want to visit the San Jacinto Monument (where Sam Houston and the Texians defeated Santa Anna after the Alamo) and Battleship Texas. The Houston Zoo and the Science Museum may be of interest.
You might have parking problems at the downtown
Aquarium Restaurant, but there's another one with better parking at the
Kemah Boardwalk near Space Center Houston.