The question was asked how we are going to go about trying to verify oral history that our Nancy Valentine Auvil was a Delaware Indian. So I will use this as an example for what I have learned from my other lines about this subject.
1. Male yDNA Haplogroups. One way to verify American Indian heritage is to have a DNA test on a male member of the family to determine their Haplogroup. The Haplogroup would be the same for the direct line of males in a family clear back to their origins. A Haplogroup of Q, C or R1 are the most commonly found in Native Americans. You can read more about it here. R1 is European and I believe it means that your ancestor inter-married with a European before coming to America or after. It is included here because a high percentage of Cherokee, for example are showing R1 yDNA Haplogroups. If your male DNA shows Q or C - that is a pretty clear indicator of your Indian heritage. If it is R1 it is not so clear but does not rule out Indian heritage. In our example of Nancy Valentine, the only way we would know is if we find a male in her direct line, a male with the surname Valentine, and we test his DNA. Women cannot be tested for yDNA. I don't know of any living male Valentines but we might find one! We should start looking.
2. Female DNA tests. If the Native American ancestor was female you can do an mtDNA test and if the test shows sub-Haplogroups of, A, B, C, D or X2a it would show ancestry indigenous to the Americas. Now, I am not sure if the maternal tests will work if the ancestor was more than 5 generations back. I'm not really sure. I am not an expert, I am just trying to summarize what I know here. For a starting point.
3. I have heard from people that if they have a maternal line that is too far back to show mtDNA heritage that there are test facilities that look for certain markers that Native Americans have that if you also have those markers it would verify your ancestry. I don't really know anything about this yet, but it would be an avenue to pursue if you could not find a male relative to test and your mtDNA is too diluted to show your heritage.
If I find out more I will post it here. I just wanted to make a little Native American Heritage 101 post.