Five days after Jason Carter jumped into the Georgia governor’s race, his grandfather, former President Jimmy Carter, delivered a speech urging a ban on the death penalty. Within hours, the newly minted candidate felt compelled to issue a retort: while he loves his grandfather, he told a reporter, “I believe in the death penalty for heinous crimes, and that won’t change when I’m governor.” The episode spoke to the benefit and potential burden of the Carter surname for the upstart Democratic state senator, who is waging a long-shot bid to unseat first-term Republican Gov. Nathan Deal. Practically everyone in the...