I can't sleep and I have nothing productive to do tonight so I looked into this Cherokee ancestry thing.
The ancestor in question, Neoma (or Oma) O.C. Sarah Smith, was almost certainly the child of Wyatt and Peggy Smith, and there's no indication either of them had Cherokee ancestry. I suppose it's possible that when the family moved from North Carolina to Tennessee and they were passing through the Cherokee nation that they took in a young Cherokee girl- her name doesn't actually show up on records until 1820, when she was married to J.H. Crawford and living next door to Wyatt and Peggy in Tennessee- but I don't know how likely something like that is to have happened. It would explain the weird name and initials possibly hiding a Cherokee name, though, in trying to pass as white.
I think it's more likely though that William J Crawford in his 1894 marriage was just putting his mom's race down as Cherokee to try to prove ancestry to get a free Indian land grant from the Dawes Commission (which was established a year before). Literally tens of thousands of whites tried to falsely prove Native American ancestry to the Commission to get a land grant.
Not saying she doesn't actually have Native American ancestry or anything; being a white person from Oklahoma she's probably got 1/32 ancestry from some relative anyway. This probably isn't the source, though.