Magic House is PERFECT for your child's age. There will be lots and lots of age-appropriate things for you to do, and it's fun for both kids and their parents.
One recommendation is if you go to Drury's hotel website, you can usually find some coupons including a buy 1 get 1 free for Magic House.
Another tip is to try to get there soon after they open if at all possible. They get VERY busy and it's not a wide open place like some other museums.
Science Center is probably too advanced for a 3 yr old.
City Museum could be fun, although maybe not a ton for her age. They do have an indoor aquarium there if she's into that kind of thing yet.
The zoo is always a great option, it's free (does have some fee based attractions as well as parking). And Grant's Farm is GREAT for her age, and FREE too.
I 100%, absolutely recommend staying at a Drury hotel. They are awesome and extremely family-friendly. No, I don't work there but we got there 3-4x a year and will not stay anywhere else. We usually can find a great rate there - $70-80 a nite.
A while back I wrote a St. Louis resource for a local website I provide content to, here is the link:
http://springfieldmoms.org/st-louis-day-trips-and-attractions/ definitely check that out for some ideas.
We love going down there, it's about 100 miles from where we live so it's a nice easy drive.
Have fun and be sure to let us know all about it!