"Cherokee" is a common family legend in the South East, much like having Wyatt Earp's illegitimate child in the family tree in Texas, Oklahoma and Kansas.

I was always taught that the claim of having a Cherokee princess in the family tree was often used to give nativism and white supremacy more credibility through self-Indigenization, which is what helped it spread and survive to the current day. And as others have pointed out, it was also used as a way to hide race mixing. It's likely that a lot of people aren't aware of this, and just think they're sharing a fun but if family trivia.

And, as I pointed out in another comment, the Cherokee Nation has no requirement for any percentage of native ancestry, so there are a lot of people in Oklahoma and the surrounding area who are more or less white, but are legit members of the Nation under it's bylaws. Which can add some confusion to the issue.