CHICAGO—Cameron Percy thought he’d found a rare treasure when he moved into a new place two weeks ago. Someone had left behind dozens of Beanie Babies in a box in the basement.

The 28-year-old took to eBay, Reddit and Instagram, and concluded the small stuffed animals, a craze from the 1990s, will one day soar in value, like the bitcoins he bought in early January.

“A...