I'm in Houston and there are a number of places you could consider, kinda depends on your price range, how much time you've got and where you're going to be located. Being from Chicagoland myself originally, I can say things are a bit more spread out down here as opposed to Chicago and the burbs. Here are a few things that come to mind...
1. There's the Aquarium downtown which has some rides, a restaurant, and of course, a pretty cool aquarium..
2. There's poker rooms, strip clubs and underground game rooms (slots) all over the place if any of those things are your cup of tea..
3. Space Center Houston..which shouldn't need much explaining, but you'll need to go during the day..
4. Several different museums if that's your thing..
5. Take a drive out to Galveston and you can visit the beach and Pleasure Pier (rides/roller coaster type stuff)
Once again, it kind of depends where you're going to be. If South of downtown then Space Center and Galveston are a short drive. If North of downtown you could be talking a much longer commute. If you're wanting to give your cousin a taste of Chicago we've got a couple of Rosati's Pizza's in town and they've got the best Italian Beef's you'll find in Houston.
I don't know anyone selling tickets to Saturday's game, but there's some on SeatGeek right now going for $585.