What it really means when they're "part Cherokee": "one of our ancestors had sex with a Black person and we're all too ashamed to admit it, so we told you you're part Cherokee to make you feel better."
Oh that's Seminole. Which most racist people are too ignorant to realize meant runway slave (In this context, the majority of Seminole were/are straight up indigenous), and thus actually a black ancestor they didn't want to acknowledge. But they had to put something on the family tree in the family bible. That's a fun one to find in the South East US.
Cherokee is usually a straight up fiction. Personally I think the idea got popularized because people like feeling special and native Americans were big news in the 1970's. Racists also jumped on it as a means of showing native Americans didn't deserve special rules or treatment.
In Oklahoma, if you can trace your ancestry back to someone who was on the Dawes Rolls, you can apply to be a member of the Cherokee Nation regardless of your percentage of native ancestry. So there are a lot of people who are effectively white, but are part of the Nation and consider themselves part Cherokee.
This is distinct from the "part Cherokee" or "descended form a Cherokee princess" claims that were used to try and legitimize white supremacy in the south.