The way it was explained to me, Columbus thought he landed in India, not a new continent. The people he met were, logically, Indians. And the misnomer stuck. American Indian was coined not to be confused with East Indian. Of late, Native American, predominately used in the USA, was invented to do away with Indian altogether. Whether this name replaces American Indian remains to be seen.
Unless the English language has changed, a native is a person born in a particular place, country, etc. I was born in California, which makes me a native Californian. According to my Webster's, this also means I am as native American as a Native American.
This also means children of illegal immigrants born in this country are also native Americans. How can anyone be sent back to their country of origin when the USA is their country of origin?