What unique characteristics of Mexico contributed to the culture of the civilizations in the region in 1750-1914?
Once you get as far forward in history as 1750 and beyond, it is very difficult to discern unique characteristics of countries or to say that they have changed the entire civilization of a region.
For example, one major contribution that Mexico made during this time was to become the first Spanish colony to break away from Spain. But this was heavily influenced by things like the American Revolution and the Enlightenment and was not uniquely Mexican. However, Mexico's rebellion did help to pave the way for other wars of independence in Latin America.
As another example, a major aspect of Mexican culture is "mestizaje," or the mix of Spanish and Indian influences. However, it is hard to say that Mexico spread this to other countries in Latin America when they, too, had cultures that were influenced by their own mixing of Indian and Spanish influences.
Finally, we can look at something like the positivism of Porfirio Diaz and the cientificos. This way of looking at society was also taken up in Brazil. But again, this was really the result of both Mexicans and Brazilians taking in ideas from Europe (especially those of Auguste Comte) separately.
Therefore, it is difficult to identify any of these things as truly unique to Mexico and as influential to the region as a whole. If I had to pick one, I would pick the fact that Mexico was the first colony to break away from Spain.