EL RETIRO, Colombia—This misty town in the rolling hills of northern Colombia is home to 47-year-old Gudiela, a raven-haired, amiable mother of two who researchers say could help unlock a cure for Alzheimer’s.

Gudiela is part of an extended family here whose members, in a centuries-old twist of genetic fate, are likely to carry a rare gene that leads to an early-onset form of the disease that can cause full-blown dementia as early as age 45.

...