In Latin America, their problems are rooted in the colonial mismanagement of the Spanish, the dominance of the Catholic Church, the power plays and the Americans' involvements in their politics. Especially that last one.
It all depends what divergeance point you are asking. If you're asking for them to rise post-1900, that's a tall order, but not impossible.
One idea could be to where the Great War ends up differently, maybe under Theodore Roosevelt and thus the situation over in Europe is different. One where the USSR and Nazi Germany does not really rise up to cause problems. This means no communist boogieman so socialism and communist along with their milder counterparts are not spooky to the mainstream. Next, we could have it to where the Spanish Civil War has many of the left flee after Franco gets involved and so on, thus fleeing to Mexico. Maybe this along with the potential Catholic Church's siding of this and maybe even their calls against leftist rhetoric would cause Catholicism to greater lose popularity within Mexico. If under good leadership and if the US leaves well enough alone, Mexico could rise up more and more to become quite prominent while serving as an influence in Latin America.
Things that would help would be distrbuting the population more equally, greater environmental protections and so on. Probably decriminalizing drugs would help too. Radical maybe, but it'd help in the long run.
