Honestly, this is gonna be a tough question to answer, but here goes: One majorly important POD that could work? Have Martin Van Buren *not* run for a second term in 1844 and have somebody like James K. Polk or Lewis Cass win instead. Polk, especially, was not interested in annexing even more than a few parts of Mexico-those claimed by the Republic of Texas prior to it's statehood in Dec. 1845-and outside of getting Texas, mainly only had his sights set on the Oregon Country(and indeed, Polk, as a key player in resolving the Oregon dispute, was one of those responsible for extending the 49th parallel border all the way up to the Puget Sound); even Pres. Van Buren was hesitant to get involved with California or the northern interior until the Mexican gov't went after the independence movement in the former area-and Polk himself was one of a number of Democrats who had originally openly opposed the annexation of all of Mexico, even going so far as to warn that such a move would backfire tremendously(it did backfire pretty significantly, but not for the reasons Polk thought it would).
As for what would happen to Mexico ITTL? Well, IOTL, after the U.S. gave up on Mexico for good during Reconstruction(more specifically, in 1875, after the end of the Second War Between the States-Lincoln hadn't initially wanted to pull out of Mexico, but the first war had gone poorly for the Union, so he didn't have a choice), President Grant *did* end up having the U.S. keep most of what was left anywhere due south of Arizona + N.M.(which had been occupied by the Confederates before the U.S. liberated them in July 1871-the Californias, both Media and Baja, had both broken away in Feb. 1872 thanks to noted filibusterer William Mahone, and while the Mexican government had hoped to re-establish control over these territories, couldn't hold on them for long, especially as the anti-Juarez faction most popular in the far north blamed him, specifically, for the fighting between him and the Confederates), as a repayment for assisting Juarez's administration in the First Mexican Civil War, with only Sinaloa, Durango, and Tamaulipas(Nuevo Leon was absorbed into both Coahuila and Tamaulipas to punish the anti-Juarez forces) returning to Mexican administration. Mexico then saw the establishment of the world's first socialist state in 1886(albeit at least it was peaceful at first), and then one last civil war, before the U.S. intervention helped re-install a more stable government after the end of the Second Mexican Civil War in February 1912. So yeah, there was a lot of carnage, both proverbial and literal, during that time.
ITTL, if the U.S. had allowed Mexico to remain independent post-1848, then that country might have had a rather less rocky transition to a truly modern democratic state; the stability that Mexico had between 1876 and 1907 was certainly welcome but things definitely. I would imagine that a socialist movement might still arise, but it might well be delayed by a few decades, at least, without 15 years' worth of occupation effectively throwing Mexico into socio-political chaos.
Edit: Presidents of Mexico 1863-1912:
1863-1870: Antonio Lopez de Santa Anna**[1]
1870-1879: Benito Juarez[2]
1879-1884: Porfirio Diaz***[3]
1884-1886: Sebastian Lerdo de Tejada[4]
1886-1897: Jose Maria Iglesias (Worker's Revolutionary Party)[5]
1897-1905: Francisco Leon De La Barra (Worker's Revolutionary Party)[6]
1905-1907: Victoriano Huerta (Worker's Revolutionary Party)*[7]
1907-1909: Francisco Carvajal (Worker's Revolutionary Party)****[8]
Second Mexican Civil War
1909-1912: Porfirio Diaz (Conservative Restoration)*****[9]
1909-1912: Roberto Carranza de la Garza (Liberal Reformists)[10]
[1]Santa Anna's comeback was unexpected to many, but his adminstration did manage to put Mexico back on track-unfortunately, more long-lasting stability didn't come until after the end of the First Mexican Civil War
[2]Still regarded as the greatest president Mexico ever had, and after the dissolution of the C.S.A. in 1874, the Mexican government helped track down numerous Confederate war criminals who'd attempted to flee southwards.
[3]Seen as the most unworthy possible successor to Juarez-deeply corrupt and removed from office-later led the rightist faction of the Second Mexican Civil War, partly as revenge against the Juarezistas and their allies.
[4]Lost the 1886 election to Iglesias and his new democratic socialist party, but later became longtime ambassador to the U.S.
[5]Was well liked by many, but by 1897 his health was in decline, so he opted to ask the People's Congress to call an election that year, to which they obliged with pver 75% of the vote in favor.
[6]Leon De La Barra decided not to run again in 1904, and supported Victoriano Huerta in that year's election, and in 1906.
[7]Sadly, was assassinated in 1907 by a member of a radical Catholic sect known as the Cruzeros.
[8]The 1908 election between him and Pedro Lascurain Paredes of the liberal-capitalist Reform Party was unexpectedly close, especially after the recession which began in May of that year-after the December election, neither man had won a majority of the vote, but Lascurain came ahead after a recount by about 5,000 votes. Carvajal, though disappointed in his loss, urged the people of Mexico to support the new administration but Lascurain's assassination in February 1909 by one of Porfirio Diaz's more fanatical backers threw the country into chaos-and with Diaz's forces attempting to take over the whole country, 35 years of stability ended with a bang, and Lascurain's running mate, Roberto Carranza, rallied a large number of supporters behind him and the Second Mexican Civil War had begun.....
[9]Porfirio Diaz ended up becoming one of the pre-eminent voices of the staunch conservative right in Mexico during the 1890s and by 1906 had amassed a fairly large devoted following, and many assumed he'd run for office again someday. But after Pedro Lascurain lambasted Diaz for his rhetoric in the fall of 1908, and then condemned his movement for attacks on pro-socialist forces throughout that winter, Diaz quickly turned to armed reaction instead. And from late 1909 until early 1911 things looked dark for the anti-Diaz forces, until U.S. President Charles Evans Hughes approved a formal intervention in February 1911; and then it was the Porfiristas who found themselves in deep trouble. One year later, with his control over Mexico having all but crumbled, Diaz's encampment was struck by an artillery shell just outside of Cd. Tlaxcala on Feb. 24th, and he died the next morning; that same day, his top generals surrendered to Carranza's forces and their U.S. allies.
[10]TTL's Jose Venustiano Carranza. After the Civil War, he would win the election of 1916, then go on to serve two terms before retiring in 1924. He then served as Ambassador to the U.S. between 1929 and 1941 before dying in Mexico City in October 1949.
*Assassinated
**Resigned
***Removed from office.
****Government dissolved
*****Died fighting against the Reformists and their American allies.
