As I am sure that you are well aware, your research question will be a challenging one to answer given such limited resources. The best place to start would be by consulting our section on "Native American Heritage" at: American Indian Records in the National Archives | National Archives
Another possible resource is provided by the National Park Service, is titled: "Honoring Tribal Legacies" and can be found here: HONORING TRIBAL LEGACIES
You might consider contacting Carol Buswell, Education Specialist at National Archives and Records Admin. In my mind, she's the expert in Native American research. She works out of National Archives at Seattle. She's also one of the 'people' that are on History Hub. You can read her profile here: Carol Buswell . She might have some suggestions for you.
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First, let's look at the information you have on the ancestor besides his/her name ...
* Where was he/she living? Were they living on a reservation?
* What is the time period?
* What does your family story say was his/her tribal association? (Does their geographic location indicate the same tribe?)
* How do you know the Native name? Was it from a document? Who wrote the document?
Depending upon the tribe you are looking for, they might be in the Indian Census under their phonetic name. Ancestry.com (free at many public libraries) has an index of the Indian Census (1885 to 1949). These census returns were usually completed every year by the BIA. Try searching the name as a first name and then a second name in two searches. To access the reservation censuses, go to Ancestry.com, click on search, click on card catalog, type Indian Census into the search box.
If not on a reservation, a regular census search for the phonetic name (one search with the name in the first name box and another with the name in the surname box) might also give you results. You might also want to sound it out in your head and try different spellings as well.