Texas Medical Center, in Houston, is the largest, and one of the most advanced, medical centers in the world.
IMO, a follow up with a specialist from any of its hospitals would be a major help to Sheldon.
Cheers!

What in the wide, wide world of sports is a-goin' on here?

Of course it's a joke, and there is no deep meaning, the obvious meaning is right on the surface. The problem comes if someone thinks the joke is that Houston is just the nearest big city, or whatever.
Yes, it's definitely a joke. And "Houston" is not the punch-line. Not even if you're thinking "Planet Houston" from the Superman movies.

And what you seem to be missing is that IT'S A F'KING JOKE! Quit looking for deeper meaning!
I knew this guy who thought there was a hidden message to Motley Crue's "Girls Girls Girls." Hidden message: the song was about GIRLS!
You know you've gone over the top when
I
have to tell you to lighten up.
Chicago Cubs
:
WORLD SERIES CHAMPIONS!!!

What you seem to be missing is that the joke/reference isn't about Houston, or about a second opinion, it's about a SPECIALIST. The idea being that a regular family doctor wouldn't be able to accurately determine what Sheldon's problem is. But since she didn't take him to see a specialist for a proper diagnosis, Sheldon uses the excuse to say he's not crazy, because "my mother had me tested."

You are WAY overthinking the joke. She regretted not getting a second opinion. There is nothing else to it. Move on. LOL
"You are a smelly pirate hooker!"
"Why don't you go back to your home on Whore Island?"

Sometimes, like in this case, it is not deeper.
I am also not American, but I do know there are more things in Houston than Nasa and space related stuff, also we can deduct from context that she means some sort of medical professional.
My guess is that she got the recommendation from her doctor to seek this specialist to come up with a diagnosis for Sheldon, which would be either a medical or psychological/psychotherapy specialist.

Frankly, most of the jokes in this series are not very deep.
Houston is the closest major city to where Sheldon was born. That's the only significance of Houston.

Sheldon's mother did mention another specialist that was recommeded by the first.

I don't think the issue of a specialist would have even come up, if not recommended by an earlier person, a doctor in this case. Typically a specialist requires a referral.

You're assuming the first doctor advised her to see another specialist. She never mentioned it.
And it's just a joke.
Chicago Cubs
:
WORLD SERIES CHAMPIONS!!!

No, that's not all there is to it. Sheesh.

You're over-thinking it. Sheldon is from Texas. Houston is a major city in Texas, so there would be a lot of medical specialists there. That's it.

This message has been deleted.

Sometimes a joke is just a joke. Besides, she DID have him tested. Now she regretting not having him tested enough after some of the crazy things he's done.
Chicago Cubs
:
WORLD SERIES CHAMPIONS!!!

Nah, it has to be deeper than that. The way jokes and references are set up in the series, its never just 'superficial', there's always a deeper reference. And the way this one is set up, clearly the joke is more the type of specialist it would have been. Now, I'm not an american, the only thing I think of when hearing houston is obviously Nasa/space related, but I don't believe that to be it.
The question is, what would that 'speciality' have been, what kind of irregular science or specialization was taking place specifically in houston in what would have been the mid to late eighties?
Without that piece of information the comment she makes is completely devoid of any point which would be completely uncharacteristic of those kind of reference-based jokes in BBT.

Evidently the first tester advised her to see a higher-level expert/specialist, but she didn't and regrets that omission.

that part i got, thanks.

It's part of a running joke whenever Sheldon was accused of being crazy, he'd respond "I'm not crazy; my mother had me tested!" Clearly she regrets not getting that second opinion.
Chicago Cubs
:
WORLD SERIES CHAMPIONS!!!