The timeline linked in my sig deals with the survival of the First Mexican Republic, complete with a Mexican victory in that TTL's Mexican-American War.
Okay plug done, but yeah the the POD I used was at the very start of the War for Independence in September 1810. The Querétaro Conspiracy originally planned to revolt in December, but once the Spanish caught wind of their plans 3 months early they forced the whole thing into motion. Now the interesting thing about the conspiracists plans was that Hidalgo wasn't meant to be the leading military commander, it was supposed to be the two military officers that were part of the conspiracy, Ignacio Allende and Juan Aldama. In the chaos and confusion the morning independence began, the charismatic Hidalgo usurped control of the insurgent army and well, the rest is history.
Apparently the three men had talked the matter over and agreed on Hidalgo...I wanna say after the insurgents met up with the Queen's Dragoons in San Miguel de Allende, but I can't remember. Anyway, later on at the Battle of Guanajuato, the insurgents (who were mostly indian peasants recruited from the region) pillage the town and kill thousands of people, which caused a rift between Allende and Hidalgo.
In my TL, Allende and Hidalgo agree to share joint command in order to remain more loyal to the original plan, and in the aftermath of what occurred in Guanajuato, Allende would become supreme commander. One month later when the insurgents reach the Valley of Mexico, Allende leads them into the capital, and despite losing a lot of his men takes Mexico City (contrast to Hidalgo's unexpected retreat after Monte de las Cruces).
Now this POD isn't the only one you can use, but it is an important one (and really early too which helps). Later POD's can also work, such as keeping Morelos from dying in 1815. After 1824 you may be able to keep the republic going if you prevent Guerrero's coup, but by the late 20's and early 30's things were already getting pretty bad. Not impossible mind you, it's possible to save the republic even after Santa Anna starts fucking shit up, but the earlier the POD, the better Mexico will be off.
Concerning Santa Anna's early demise, he received an arrow wound in his left arm in 1811...had that arrow veered a little to the right it could have been a fatal injury.