How did the role of everyday Mexicans in Mexico change from the 1960s to present?
One way to answer this is to say that everyday Mexicans have gradually taken a larger role in Mexican public life beginning in the late 1960s.
For a long time, everyday Mexicans had little power. The PRI essentially controlled everything. But this started to change in the 1960s. The Tlatelolco Massacre of 1968 was a step in this process as it made people have a much more negative view of the government. The process took a huge step in 1985 when a massive earthquake hit Mexico City. The government did a terrible job of responding to it and individual Mexicans became very important in the rescue and cleanup effort. A final major step was the election of Vicente Fox as president in 2000. He was the first non-PRI president since 1929. His election showed that the people had become much more important than ever before.
In these ways, everyday Mexicans have become much more important in Mexican life in the time period you mention.