The thing that most posts like this suffer from is logistics, logistics, logistics.
Sure, Lincoln offered 50,000 troops. It's incredibly salient to ask, Where are they coming from? When Lincoln made the offer, he offered what he didn't have. Both Lincoln and Houston knew this. But for the sake of the OP, let's say that Lincoln takes Houston up on it. Cool. That doesn't happen in a vacuum. Throughout 1860 and 1861 various militia companies had formed (and were continuing to form) for service in the Confederate army.
Back to the question at hand. Houston accepts the offer from Lincoln. Given that other than Houston, the entire state government was mostly secessionists, the next thing is he gets deposed. Now, I can't tell you whether the secessionists would simply put him under house arrest or if they would go further than that.
So, it's the beginning of March, Houston is gone as Governor, Edward Clark, is now Governor. General Twiggs surrendered the all of the Federal property in Texas on February two weeks earlier, in the last days of the Buchanan presidency. Texas' military forces are mobilizing.
What are the Federal amphibious abilities in April and May of 1861? I'm not sure but safe to say, minimal. 50,000 is a pipe dream. Maybe 2 or 3,000 men could be landed at Galveston, and that would come at a trade off. And for what? Assume that Lincoln is able to muster a brigade of infantry and marines, and lands them in Galveston or on the mainland of the Texas coast in May or June 1861. They could conceivably deny the Confederacy the port of Galveston throughout the war. That's actually a very likely outcome of any attempt by Lincoln to prop up Houston. And that's not exactly a small victory in and of itself.
But on the mainland of Texas, whatever forces Lincoln could assemble would be a tiny fraction of the 50,000 promised troops. And generals like McCulloch would very likely have tactical superiority. For what it's worth, I think Lincoln would know this and would likely not act on his promise, at least by sea in the first year of the war.
I can hear someone else saying, well he could send them down from the north. See my first sentence. Logistics makes that highly unlikely. There's no viable railroads between St. Louis in Missouri and Texas. Not of any length that would be valuable for transport or supplies. They would need to cover 650 miles, over land through hostile terrain. Now 50,000 troops could likely do it, but where is Lincoln going to find 50,000 that he can afford to send gallivanting half way across the continent? What about a smaller number? In 1861 there's simply not a lot of options. I would argue even an entire Union army corps (20,000+ men) going from Saint Louis would likely find that even if the Confederate Department of the Transmississippi wasn't able to muster enough men in one place to stop them (not a sure thing), it would take half the fighting force of the Feds to hold their supply lines, otherwise, they'd been at the end of a very long tether without supplies.
Ok, what about all the Germans and Poles in Texas who opposed secession. After all, 25% of the state voted against secession. That's actually a very good question to ask, IMO. After all, 2,000 (mostly foreign born) Texans served in the Federal army. There are two things that work against this number growing to 5 or 10,000. First, while many Germans were unionists, and a few served in the Union, a few more tried and were hanged for treason by the Texas government. More than that, just because people were ardent unionists, does not follow that they will march off to war against their neighbors. Houston's deposed. Any attempt by Lincoln to relieve him by sea is simply too small in the first year of the war to be particularly successful, just a few thousand at most. Same for an overland route. A few who want adventure will take the risk, but for the vast majority of family men, the risk simply outweighs the reward.
Now, if Lincoln makes an effort, while I think it would come too late to affect the first year of the war in Texas, it will have serious repercussions in that the Confederate army of the Tennessee will have even greater manpower shortages, because Lubbock (governor in Texas from the end of 1861 to 1863) will fight tooth and nail to keep more of the 70,000+ Texans who served in the Confederate army home to oppose invasion efforts. So, and this is really playing fast and loose, if there were substantially fewer Confederates in the Western Theater (Tennessee and deep south) then the war could go South (sorry for the pun) for the south much sooner, and result in the Union cutting the south in two even earlier than happened IOTL. So, even though I think Lincoln's offer was not capable of propping up Houston or keeping Texas in the Union in 1861 or even 1862, it could potentially shorten the war by keeping tens of thousands of fighting men close to a thousand miles west of the main theaters of conflict.
Now before anyone comes in here channeling the ghost of TFSmith, keep in mind, we're talking about the facts and circumstances that existed in 1861, not 1865.