Kevin Rose, the Digg founder and Silicon Valley wunderkind, remembers the pushback he got in young-tech circles when he started wearing his late father’s Rolex a few years ago.

In a culture of T-shirts and app-centric smartphones, a mechanical Rolex Datejust from the 1980s seemed like the equivalent of gray hair on a Google intern. One web developer chided him, saying, “That’s an old-man thing,” Mr. Rose, 38, recalled.

Rather than feeling ashamed, Mr. Rose developed a fascination with analog watches. And the deeper he dived into watch collecting, the more he saw business opportunities.

“As an entrepreneur you start looking for industries that are relatively untouched by technology, making them open for new innovators to come into the space,” Mr. Rose said.