I know this is a stupid question but, why don't they fly to their destination and have the emergency landing there? If it all works out the people are where they wanted to be and if it dosen't then they are screwed anyway.
They don't want to raise the gear in case the blown tire caused other problems that would prevent the gear from lowering again. At the speeds and altitudes they'd have to use with gear down, they most likely don't have the range to get to MSP. Also, IAH is Continental's main hub, and they'd probably much rather have an airplane with maintenance problems land there than at an out-station like MSP.
Note to self. Next time read the thread to see if someone else asks the "stupid question" first. :-)
In addition to the issues mentioned, another factor is that if something else goes wrong en route, they may not be able to get to an airport as well equipped as the one they took off from. If they had taken off from a less well-equipped airport, it might have made sense to fly to a better one but when an airport with good emergency gear is in range, it's best to stick with that.