When tech startup Gnosis raised about $12.5 million in April, the company didn’t tap the typical Silicon Valley funding network of venture-capital firms or wealthy investors. Instead, it auctioned tokens online in a process called an “initial coin offering.” They sold out in 12 minutes.

These digital coins are essentially clones of the virtual currency bitcoin, which has come under scrutiny for its use in last week’s global hacking attack. The newer coins are the proverbial next big thing in the world of cryptocurrencies,...