Texas Burger Palace at 5757 Ranchester Dr #1700, Houston, TX used to be a really good place, but I haven't been there in over 20 years. They're still in business, so they must still be pretty good.
Bubba's Texas Burger Shack at 5230 Westpark Dr, Houston, TX is long known for their great burgers.
I think someone already mentioned Lankford's Grocery & Market 88 Dennis St, Houston, TX. I've never been, but the Houston TWT Tuesday breakfast is there some weeks. It's cash only.
I've heard good things about Stanton's City Bites 1420 Edwards St, Houston, TX. Mish Mish on here said she wants to try it out.
Used to go to Roznovsky's Hamburgers 3401 W T C Jester Blvd, Houston, TX weekly back in the '90s and they had really good burgers with the freshest condiments. Haven't been there in years and the interwebz says they're permanently closed.
Guy's Meat Market 3106 Old Spanish Trail, Houston, TX used to serve a limited amount of burgers starting around 11:30 Monday - Friday. The line was usually out the door and if you got there after 12:00, you probably weren't going to get one. It's looks like they've remodeled and do regular BBQ meals there as well now. They're open 9:30 - 5:30 so I guess the noon rush is no longer an issue.
Miller's Cafe is one of the regular stops in rotation for the Wednesday Clear Lake Bike Night. Burgers done very well most any way you can imagine. They have 4 locations around Houston; we go to the one at Space Center and Bay Area Blvd. Very similar to Texas Burger Palace.
http://www.millerscafehouston.com/locations.html
There's always the higher end chains - Beck's Prime, Smash Burger, Five Guys, Red Robin, and there's at least one Prince's still in business off the Katy Freeway between Dairy Ashford and Kirkwood on the south side. Prince's was THE Houston drive-in burger stand back in the '50s & '60s.
http://www.princeshamburgers.com/?page=history
I'm sure there are a lot more around the Houston area I'm not familiar with, but always ready for a good burger - probably should have been named Wimpy.