Here's the challenge -- what should I absolutely do while I'm in Houston for one day and one night? Suggestions that are close to the Galleria are welcome since I'm staying at a hotel there (St. Regis) and will not have a car. Thought it sounds like I'll be taking taxi's. Best whatever you can recommend is most welcome!

Ann - Stay in the room and catch up on some pay-per-view movies...sleep... and then leave. Houston is one of the most disapointing cities for travelers that I have ever been to. If you are not interested in Men's Strip Clubs, get out of there as fast as you can...

Choice 2 - rent a car and drive to Galveston...

I would agree there's not too many must-do's; but a couple of things that Ii could recommend ( I've spent many weeks in Houston for business-a NYer here)
See if you can see a performance at the Wortham Center downtown, The Houston Opera, or a concert at Bayou Place. There are a couple of nice jazz clubs at Bayou Place, which also downtown. I'm particularly found of Sambucca's, also downtown, which is a nice restaurant specializing in seafood, and there is live jazz performed in the evening.
The Museum District is nice to stroll around, although in match for NY's museum or the Art Institute in Chicago.
Not too much more to do given your time and transportation options.

First, you need to rent a car as Houston is for driving only. I would recommend that you go south on I-45 about 30 minutes to NASA Road 1 and go left to NASA. It is a great facility in a very nice area. There are plenty of nice eating places there and further east in Kemah on the Boardwalk. Go back to I-45 and south to Galveston where you can stroll the Strand, see historic houses and ride a ferry & feed the seagulls. If you eat in Galveston, go to Gaido's (near the beach) or Clary's (behind the Galveston Daily News on Teichlemann Road just across the causeway).
If you stay in Houston, go to the Alley Theater or the Symphony or the museums. There are many good eating places.

Poor Houston, someone always has to pick on it. (though I always suspect those people haven't given it a chance)Seriously, it's an underrated city, especially if you like museums. The Museum of Fine Arts Houston has recently opened a big addition; the Menil Collection has the coolest collection of surrealist art, and the campus also has the Rothko Chapel, very interesting building; and the Art Car Museum and the Orange Show, this very weird folk art installation, are like nothing else.
IF you possibly can, eat at Cafe Annie _ best upscale Southwestern anywhere, including the mansion on turtle creek.
And yes, if at all possible, rent a car _ Houston is much easier if you have your own wheels.