The clearest answer is that Europeans generally referred to the whole of East and South Asia as the 'Indies'. So, once Columbus landed in Caribbean, he referred to its people as "Indians." Even once the mistake was realized, the accepted explanation goes, the name still stuck. Since the tribes held such widely varying traditions, languages, and geographic placements, it was not as easy to personalize nomenclature and ensure consistency amongst the settling and exploring Europeans. It was far easier to generalize them as the name given to them at the start "Indians." A comparable example is the lumping of all citizens of Spain as "Spaniards." Spain itself is broken into many regions which refer to themselves first and foremost by that region (some have even objected in the past of being under the territory of the collective 'Spain').

Another interesting, but almost certainly false, theory I know of is that the name "Indian" actually is derived from the Latin "In Dios." This google ebook gives a great explanation of the idea. The author himself refutes the idea, as the most common piece of evidence (that India was known as Hindustan and so Columbus did not believe he had landed in present India at all) is false since Hindustan does not enter common vocabulary until the 17th century.

The above theory was popularized as early as 1984 by Peter Mathiessen's book, Indian Country, and the comedian George Carlin. Besides teaching us that we have to be wary of the histories we read, it demonstrates the difficulty of a obtaining a single answer in any historical topic.