Growing up in school we studied Christopher Columbus many times, and it seemed like in each year we learned a contradictory version of the story. So I really have no idea why the Indians were called Indians.
One year we learned that Columbus called them Indians because he had thought he had gone to India. Another year we learned that Columbus had actually thought he was in the West Indies, and not India. Finally, one time we had a school assembly where some actual Indians (Cherokee, I think) came and gave a presentation about their culture and their history, and they said that Columbus didn't call them Indians because he was confused about where he was, but rather as an abbreviation for "Indiosis People" (I have no idea the actual spelling for this word, as I only heard it, and I didn't write it down), which meant "Beautiful People" in some language, but I don't remember which one.
As a child these conflicting stories really confused me, and even today I don't know which one is true. I've usually tended to believe the version the Indians who came to our school told us, seeing as they were actual Indians, I think they would know. But I've never heard of the word "Indiosis" after that, and have never been able to find it anywhere, but that may not mean anything because I don't know the spelling.
The first theory, that Columbus thought he had reached India: This is widely believed and has been taught to American schoolchildren for generations. Columbus actually thought he had reached the East Indies, also known as The Spice Islands, which is the part of the world he was aiming for when he left Spain. Frankly, European knowledge of the Asian continent was quite limited, and the entire area was often vaguely referred to as 'The Indies'. The East Indies included islands that are now parts of Indonesia, Malaysia and The Philippines. As to how quickly Columbus realized he was in the wrong place, people are still arguing over that.
The second story doesn't make any sense. The term 'West Indies' is an old term for the Caribbean Islands, which is where Columbus initially landed. It was used to differentiate them from the East Indies (once everyone realized Columbus had made a massive miscalculation.)
The third story I have never heard of. I wonder if they were talking about the En Dios theory? En Dios means 'of God' in Spanish. A number of people, including several Native American writers, have claimed that Columbus wrote in his journal that the indigenous people he encountered were 'people of God' and that this was eventually shortened to Indios, the Spanish word for Indians. However this does not appear in any of Columbus' writings and is considered a false etymology by scholars.