Hispanicized Names for Indians

Does anyone know how the Indians of Mexico were given their Spanish names and how far back the custom started? (After thinking about it I guess it started from the beginning. Recall that St. Juan Diego was a pre-Columbian native who had an Indian name.)

Were they given names, much like the black slaves in the U.S. by people they worked for or were indentured to? Were their native names changed at baptism? Were they given the surnames of influential people in the area where the resided?

Does anyone have a record that shows this process?

Are there any references that might answer these questions?

Thank you.

Michael Carrillo