While I don't know your arrival & departure times, I hesitate to recommend the complex and time consuming auto rental and venturing far afield from IAH.

We've developeda regular routine for a similar evolution...(driving down from Waco, 180 mi., to catch a morning flight, planning to arrive about 7PM...

Check into Springhill Suites, almost brand new, only a coupe of miles from the IAH terminals, and with the best (free) shuttle servide I've ever seen. The Springhill (a marriott "brand" with a low reward redemption level) actually has a bar, well-patronized in the evening by business travelers, often oil/gas/petrochem related, a decent breakfast "buffet' of slightly better than subsistence quality "motel" offerings, a lots of marginal cable TV channels, free Internet, IIRC, and offers a bonus, "free parking" for extended periods (of no use to you, but a reward for us).

Pick up a sandwich at one of the nearby "Fast Food' outlets, even Subway. It ain't much, but beats driving across Houston, which sprawls across the Coastal Plain. Relax, because tomorrow will tire you mors than you think.