I think the new Paleontology exhibit at the HMNS is really cool.
Discovery Green Park can also be fun. It has a few play areas for kids, a lake where you can rent remote control boats and/or sailboats for your kid to drive, restaurants in the park and a lot very close by. There's also an amphitheater but I don't know what is currently playing, if anything at all.
There is also NASA but that can be a bit of a drive if you wanted to stay close to downtown.
Sorry, I've lived in Houston my entire life and I feel like I should know more but I don't have kids and haven't lived downtown in years. Enjoy your trip!