I have a 2-part question:
-what would it take to have the Republic of the Rio Grande succeed in independence from Mexico?
*I was thinking that perhaps a greater number of empresario grants for colonization from the Mexican government caused a larger Anglo population flow, and some Texans coming into the area as well, could help this out.
-if the RRG had joined the United States, after having gained independence, what would the state be named?
The problem here is that the Rio Grande Rep (or Republica del Rio Bravo) had nothing to do with Texas' independence. It actually happened later in the years between the Texas Revolution (or independence) and the Mex-American war during the First French Intervention (AKA the Pastry War). The movement was more of a way of sticking it up to Santa Anna, who had just lost Texas, placed the country in huge debt, and caused a war against France, than actually striving for independence. There were also no Anglos in the seceding provinces.
Two ways of having the Republic actually achieve independence would be to have either war be much more devastating.
If the US declares war against Mexico during the Texas revolution (maybe Santa Anna crosses the Sabine, and Andrew Jackson backs Houston as promised) then thing would be much worse for Mexico in the early 1940s (hard to imagine but possible) and the movement might become a legit attempt towards secession.
The other one would be to have Zaragoza defeated in Puebla during the Pastry War the French reaching Mexico City and forcing Santa Anna to accept their terms. Once again the country would be in worse shape, Santa Anna would be in no shape to stop the movement at the same time the capital and Veracruz are under French occupation.
Finally if the Rep of Rio Grande actually achieves independence it will face a huge problem. The Nueces Strip would be a disputed area between it and Texas (or the USA depending on the situation) and its capital, Laredo, is actually located in this territory. Rio Grande vs Texas might be a fair match, but Rio Grande vs. the USA would be a terrible situation, that will probably end in full annexation to the US, or it begging Mexico to be taken back.
So the answer to the first one is, Yes, it could achieve independence but having this little country survive would actually quite difficult.
If the Republic is annexed by the US, it would have to become more than one state, after all the nation was made of 3 separate states. Like Texas it is likely to be given a deal, of having the option to be split into several states. And the names of the states would be Coahuila, New Leon (Nuevo Leon), and Tamaulipas.