Caleb Hanna was a child when Barack Obama won the White House and, like many African Americans, he recalled his excitement at seeing “someone who looks like me become president.” But Mr. Hanna, now 19, soured on Mr. Obama’s policies. This month, he was elected to the West Virginia House of Delegates as a Republican, becoming one of the party’s youngest black legislators in the country. Mr. Hanna grew up not only in a state that is overwhelmingly white, but in a white family. He was adopted as a child, and his father, mother, four sisters and brother, who are...