Don't stay IN Jackson PLEASE! Try going out toward Lakeland Drive (Flowood MS). The museums are out that way. I think they still have the dinosaur exhibit going on? If you or your child likes baseball, the Pearl Braves have a GREAT stadium right off the interstate, near Bass Pro Shop, in Pearl. I like the zoo, but the area around it can be kinda scary
If you are staying awhile, there is a wonderful waterpark in Philadelphia MS-Geyser Falls-run by our local band of Choctaw Indians. It would make for a fun day (open from 10:30 until 6). We have season passes and go once a week-would go more if gas wasn't so high!
Closer to Jackson, down I20, Pelahatchie has a Yogi Bear park with water areas and play grounds for kids. They also have cabins to rent.
Roosevelt State Park in Morton (right off I-20 exit 77) has great cabins, waterslide and pool, nature trails, deer pens, etc. The cabins are around $80 a night, with a kitchen, bedroom, and bathroom. They have screen porches and are on the lake. Very rustic, but very peaceful-just depends on what you like.
I haven't been to Vicksburg in a long time, but remember going to some Civil War museums while in school.