Whether someone is a Cherokee seems a question of social and personal identity. I believe that strictly from a statistical-genetic perspective, the most we can do is to compare genes and give a probabilistic value of somebody belonging to a certain genetic pool. Just consider mutations and any inter-race breeding that has happened during milleniums to highlight the fuzziness of the notion of defining somebody's race in any exact quantifiable terms. In addition, the notion of being 1/2exactly 1/2 Cherokee from statistical-genetic perspective does not make sense since the distribution of genes is not 50%:50% as very insightfully pointed out in the comments by @SteveJessop. There seems to be no mathematically sensical way to say somebody is 100% or 0% Cherokee, even less 1/n Cherokees for any choice of integer value n > 1, by just looking at their genetic markup.
Thus I would say the definition of being a proud member of Cherokee is a matter of social and personal identification. This definition is gray, but at least it gives us sample members of Human Race that we can meaningfully call fully Cherokee or totally not being a Cherokee (while ignoring the grey area cases), which validates the use of fractions in defining somebody's degree of "Cherokeeness".
After thus laying foundational definitions, we refer to the curious cases of children with three parents 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, and in a sense have three genetic parents. Let's assume one of the parents of such a child would be a Cherokee and two others "totally not Cherokees", and furthermore to make the case stronger, let's assume all three genetic parents would also participate in the upbringing and support of the child so that there really would be three parents in the strongest imaginable and possible sense. We could argue (though not with any mathematical rigor) that the resulting offspring would be 1/3 Cherokee. Now if she or he would produce an offspring, it would make some sense (even if not mathematically very rigorous) to say that offspring would be 1/6 Cherokee. Paring 1/6 Cherokee with non-Cherokee parent would give us an offspring that might want to call himself/herself 1/12 Cherokee.
Logically speaking, though, the mentioned case cannot be 1/12 Cherokee by this avenue because this controversial treatment option has not been in existence long enough for any such three parent child to have a grandchild.