I recommend the Missouri Institute of Natural Science. It's a great little museum with lots of fossils and minerals to look at. There's even dinosaur bones (from Wyoming, not Missouri) that are super cool.

Edited to add: it's also a free museum, so there is no admission.

There's a great natural science museum, the Missouri Institute of Natural Science that's really fun to go walk around and learn. Also some trails nearby for some fresh air

Potters house is usually open late. Good place to study near MSU

If they are a fan of dinosaurs: the Missouri Institute of Natural Science! They'll rent you out space under Henry the Triceratops for an afternoon and give you a tour, have an activity for you (age appropriate) and even give the birthday kid a gift!