Blair and James are trying to start a family. Like many parents, they hope their future offspring will be healthy. They’d also like the baby to have blue eyes.

The couple, both 35, describe themselves as type-A personalities who research everything. When they decided to try for a baby, they looked into DNA testing to rule out disease-causing genetic mutations they might pass along to their child. Then they learned about a test that might help predict a future baby’s eye color.

Blue...