Churchill Had Iroquois Ancestors
I don't know where the Cherokee claim came from, but the Iroquois claim apparently comes from this:
Unfortunately for the rumor-mongers, the posting describes Anna Baker's birth to her parents Joseph and Experience Martin:
Once again, this is how Indian cultures get appropriated and assimilated into the mainstream. If everyone's an Indian, there's no reason to recognize tribal sovereignty or separate Indian nations. Indians can forfeit their land and treaty rights and disappear into the "melting pot."
For more on identity issues, see Law Students Commit "Ethnic Fraud" and Red Shirt vs. Gover.