The Houston Museum of Natural Science (HMNS) is free on Tuesday afternoons after 2pm, right when the heat in the office starts to get unbearable. You can see dinosaur bones, minerals, shells, animals, and artifacts from all over the world. Other exhibits are available for a charge, but you don't need to see the Titanic to stay cool.
The Museum of Fine Arts Houston (MFAH) is free all day on Thursdays, and is even bigger. You can wander two buildings worth of galleries, including both permanent and traveling exhibitions. Be sure to check out the amazing animated project 'City Glow' by Chiho Aoshima. It's downstairs next to Cafe Express.
The Health Museum is free on Thursdays from 2pm - 5pm. We haven't been there yet, but it sounds pretty cool. You can also check out the Children's Museum for free from 5pm - 8pm on Thursdays. We've been there with young relatives before, and recommend plenty of hand sanitizer, earplugs, and