When Columbus set off on his famous voyage of 1492, he carried with him a passport from the Spanish monarch which permitted him to travel ad partes Indie (to the regions of India), as the entire Asian landmass was referred to in those times. Columbus was particularly interested in pioneering a faster route to Japan, and all the riches that could be found there.
Columbus proposed a Westward route to Japan which was, by his reckoning, 2,300 miles West of the European landmass. The correct figure is in fact over 12,200 miles - not to mention the significant obstacle posed by the entire North and South American landmasses, which neither Columbus nor the leading scientists of the era knew anything about.
Thus, when Columbus arrived in the Antilles, after a relatively short 5-week voyage from Gran Canaria, he believed that he had reached 'India', and that the peaceful, brown-skinned natives were, by inference, Indians. Although his error became clear within decades, the name persisted, even making it into place names such as The West Indies, and Cartagena de Indias. These days the preferred term for the many and varied races of the Americas is 'indigenous people,' but the name 'Indian' is still widely used - often in a derogatory sense. Be sure not to fall into this trap on your South American adventure.
(As an aside, for a time there was an alternative theory that the term was a mis-spelt contraction of 'una gente en dios' ('a people in God') but this has long since been disproved).
Both of the photos for this post were taken by Andrew Dare in the Choquechaca Valley. Choquechaca and SA expeditions have a very special bond. Read all about it here.