You can try Gedmatch.com, when it is up again in Dec. It's algorithm is much more sensitive to small amounts of Native American DNA.
Oneproblem is that many tribes, e.g., Cherokee, were heavily admixed by the 1800s, when most were moved West. I have seem some on the Dawes Roll with 1/32 blood quantum and this is over 100 years ago. One of her parents may have been a member of a tribe. However, there was probably no way to know what % pre-Columbian Native American DNA he/she was.
Stories of N.I. heritage are quite common among many, especially Southern, families. One need only look at the rejected Cherokee claims to see how little substantiation there was for most of them, even at the time.
Since it was usually white males marrying N. I. women, rather than the other way around, you can try running an mt-DNA test on a male or female whose mother was descended through an all-female line from her. An N.I. haplogroup would be proof of the heritage. However, lack of one would not disprove it.