On one of these links they talk about the different spellings of Abenaki-it can be spelled Abnaki Wabanaki Abanaki etc. It means Eastern People or People of the dawn. The name and tribe is mainly found in Northern New England and Eastern Canada but one of the articles says at one time it was used to describe all the Eastern Algonquin speaking coastal people from Newfoundland to Virginia. If you know where your great great great grandmothers were born you might be able to look at a tribal map and figure out which Algonquin speaking tribes lived near there. There was also some movement out of New England to the Ohio River and into Oklahoma Territory after colonial times. Your relatives may have been part of a dispersal from the NE. There are also some geneological links below. One of the articles I skimmed mentions forced sterilizations of Abenaki people, that could further explain your 1/16th heritage-people were trying to survive by assimilating and intermarrying. That also bodes well for your possibilities of establishing a tie if you can find the right people to connect to. If there was forced sterilization in recent times folks are likely to welcome people of Abenaki descent finding their way back into the fold.
The Abenaki's fought fiercely to retain their lands-they have a very interesting history based on my quick skimming. I hope you will want to learn more regardless of its impact on your application.
http://faculty.marianopolis.edu/c.be...ia/abenaki.htm http://www.native-languages.org/abenaki.htm http://en.wikipedia.org/wiki/Abenaki http://www.tolatsga.org/aben.html http://www.accessgenealogy.com/nativ...lina/index.htm