In 1965, when her 10-month-old son, David, started having seizures, Joan Stokes's excitement at being a first-time mother gave way to terror. "I couldn't imagine what was wrong," she said. The pediatrician was equally baffled. David's condition was not a result of a bacterial infection — it failed to respond to antibiotics — and tests for an array of common genetic disorders came back negative. It was not until Ms. Stokes began discussing David's illness with her mother, her cousins and other relatives that she realized she belonged to a seemingly cursed lineage. "I had a brother that died in...