Think of them as the metal detectors of the digital age. Instead of combing beaches for gold, these experts use everything from supercomputers to hypnotherapy to reunite people with their lost cryptocurrency.

According to Chainalysis, a blockchain-analysis firm based in New York City, about a fifth of all bitcoin—around $20 billion—is lost, most of it permanently. When people purchase bitcoin, the details of their transactions are stored in software called a “wallet,” which they can unlock with a personal-identification number....