I like the idea of a lonely Mexican traveller stumbling on Californian gold in the late 1830s. A Mexican Gold Rush in 1838? I could see that as spurring American immigration into Mexico in California and even possibly staying there for the time being. (The idea of Irish immigration to Mexico is also an interesting idea...)
Texas becomes its own nation, albeit a satellite of the United States. For sake of butterflies, say Lewis Cass is elected in 1844. Texas and Mexico still have an agreement that it's independence will be respect required that Texas remains an independent state. President Cass annexes Texas, the territory between Neuces and the Rio Grande is disputed. President Cass sends troops to the territory, Mexican troops fire upon the Americans, and the war ensues.
Britain has been supplying and trading the Mexicans this whole time given Mexico's new found wealth. (For sake of argument, Santa Anna could stick to his earlier belief that he's retired unless his country needs him, so he's solely a military leader and not a quasi-European monarch) The Mexican Army sticks it out.
The war plays out very differently. As the USN sails through the Caribbean trying to land in Mexico, the British refuse to get directly involved, so Mexico enlists the help of Spain. From Cuba, the withering Spanish Navy (supplied by Britain nonetheless) manages to defeat the Americans.
On land, the efforts of John Fremont are put to rest as a counter-strike through the Rocky Mountains (with assistance of the Mormons in Utah) send him back to St. Louis. Meanwhile, Santa Anna's army runs up the Rio Grande Valley, securing the region for Mexico. Once General Scott's army is safely garrisoned in Galveston, Santa Anna strikes into the agreed border of the State of Texas, pillaging and striking down the morale of the American army before returning to Mexico.
1846. The Allies (Mexico, Spain, and Britain) are victorious. Britain, who was not technically a belligerent, manages the peace agreement in London. Spain is given assurance that Cuba's colonial status will not be violated. The Rio Grande Valley is given sovereignty to Mexico. Reparations are paid to Mexico and Spain for the damage done to Cuba and Coahuila.
As for the American political scene, two voices have been standing out strongly against the war in Mexico: the Whig Party's own Abraham Lincoln and Robert Toombs. Lincoln persistently demanded to know the exact spot where American blood was shed on American soil. Toombs quickly followed with a poorly carried out expansionist attempt by the "imperialists." In 1848, Lincoln/Toombs beat Buchanan/Polk in a landslide.
Obvious that slavery could expand no more, the Civil War is averted as Lincoln and Toombs agree to begin manumission of the slaves. This causes distress among the slave states, and a Compromise of 1850 which sets the date to begin manumission at 1860 and it's (supposing) end at 1870.
The Whig Party will most likely survive for quite sometime. The Democratic Party... I doubt it. Mexico, however, will most likely become the hot spot for further European immigration. Especially German, Italian, and maybe even Polish and Jewish immigration. The immigration, increased area, and newfound wealth will probably put Mexico ahead of the United States by 1914 as the dominant nation in North America. Maybe a Mexican superpower after that?
That's all I got.