Traitorfish
The Tighnahulish Kid
Common misconception, actually- it is now known that "America" is named after the Welsh businessman Richard Amerike who funded John Cabot's voyage to Newfoundland in 1497- Cabot would have been expected to name any discoveries after his financier. Originally, it only applied to the areas which Cabot actually explored, but it eventually came to be used for the entirety of both continents. Essentially, "America" is the feminine Latin form of "Amerike Land".The joke is, of course, that calling people of South and North America "Americans" is silly. In that "America" comes from the explorer Amerigo Vespucci. So, if we wanted to be properly named after the European who opened up the Western Hemisphere to European colonization, we'd all be Colombians.
The great irony of the phrase "Native American" is that acknowledges their position as the native peoples, but still forces a foreign name upon them. Still, as Scy12 said, if you're going to use a blanket term for an entire race of people, then it's only fair that it's in the one language which they share- English (at least in the context of those tribes from the modern USA and Canada).