NARA has a standard answer for Indian genealogy information but it sounds like you don't have much to start with. Without knowing the time period of your ancestor, their name, or if they were a recognized member of a tribe it's unlikely NARA has anything that places them in a Native American context.
Might I recommend one of the many fee-based DNA services? I've used one such service to break down my ancestry. Investigate them online and see what looks good for you.
If that's not your style, below is a standard answer for general research into individuals, which may lead you to discover information to make future research easier:
The National Contact Center, part of the General Services Administration, may be of help you. The Center has links to military locators and information about finding private individuals from the Social Security Administration, the Internal Revenue Service, the Department of State, and the Salvation Army.
National Contact Center (GSA)
Books and articles on locating private individuals:
Hinckley, Kathleen W.
"Locating the living: Twentieth Century Research Methodology."
National Genealogical Society Quarterly, 77 (September 1989): 186 -196.
Martin, Amy Suzanne.
"Playing Detective: How Government Records and the Freedom of Information Act Can Help You Locate a Missing Person."
Heritage Quest 7 (July-August 1991): 7-8
start and trace you family history with the censuses on Ancestry.com, that'll help show roughly where they were and if they were close to a reservation. then with those names you can start to search the indian censuses also on ancestry. then if it's true and they were in a tribe go to a nara office that has that tribe and learn more.
I can not afford to go on the sites that charge a fee. I am on some free sites. I do not know how to tell if it was near a reservation.
Someone once told me there was a place where you could get a free blood test, that would tell you, is that true??
oh I never pay for those sites either, my wife used to but used up everything she needed so now if we need to look someone up we just go local library down here that has access and I play around. ancestry is so good because they got so many of the different indian censuses on there.
don't know much about the blood testing, i'd guess it might only tell you generally if you got indian blood but i can't imagine it could tell you the tribe. especially with tribes that has different bands and you'd want to learn that to learn more abotu the rich history. maybe worth a shot? i dunno.
heres a good map of rez's across the country. https://www.nps.gov/nagpra/documents/resmap.htm
I'm unaware of any place that does free blood tests. Most services charge around $100 so I suppose it's about priorities.
Vernon makes a good point about Ancestry. Some historical societies or other public libraries have subscriptions to genealogy sites and census records are invaluable. Without you knowing which ancestor was Indian you'd have to work backwards from your known family, essentially checking out everyone.