Wikipedia to the help: http://en.wikipedia.org/wiki/Three-parent_baby:
Three-parent babies are human offspring with three genetic parents, created through a specialized form of In vitro fertilisation in which the future baby's mitochondrial DNA comes from a third party.
See also http://www.bbc.com/news/magazine-28986843 for the young case of Mrs. Saarinen.
Such a child would have DNA from three people. Assuming even distribution of DNA from the three parents, and black-whiteness of the property of being Cherokee ("imagine a perfect sphere") we could take 100% Cherokee and two 0% Cherokees, to produce a 1/3 Cherokee. Her/his offspring would be 1/6 Cherokee and subsequent offsprings 1/12 Cherokees provided later parents were 0% non-Cherokees.
All this is assuming rather loose-minded mathematical reasoning or simplistic reduction of complexities of genetics. Also, as this controversial treatment is rather new, to strictly answer the question, the self-claimed 1/12 cherokee had to have some other avenue to his stated fractional Cherokee legacy.