An idea that gets thrown around a lot (by myself included) is an independent Texas. But how likely is this?
Maybe Sam Houston dies of his war wounds in early 1836, leading more nationalist people to take over? Yet even with a guaranteed British alliance, could they ever hope to overcome their massive dept?
Lamar's faction taking over has a good chance of happening even with Houston alive.
First thing is you have to do something about the election of 1844 so Texas' application for statehood gets denied.
They will be in debt for sometime. Their first major chance to overcome that will be during a US Civil War. If Texas doesn't join the Union, this means slavery can't expand westward and the pot could possibly boil over in the late 1850's.
Texas as a neutral power bordering the Confederacy stands to make a fortune selling Southern cotton and running arms to the south. Galveston will certainly be booming.
If Britain is allied or even very friendly to Texas, the Union really won't be able to do much about this, without provoking a war with Britain. A smaller Confederacy likely means the war goes in favor of the North more quickly, but that's a whole other topic.
The brief economic boom will see Texas' creditors with some payment much to their pleasure and have invaluable capital flowing into the fledging republic.
If Texas can make it to 1880 or so still independent, they're pretty much assured to last into modern times when the oil boom hits. Of course there's likely to be another war with Mexico, though British intervention may head that off.
Even in the most optimistic Texas scenarios I don't see the Texas Republic stretching from the Gulf of Mexico to the Pacific as Lamar envisioned.