Fidelity Investments Chief Executive Abigail Johnson, the granddaughter of the financial behemoth’s founder, checks the pulse of the investing world from an unlikely place these days: Reddit’s stock-picking forums.

Not long ago, Fidelity appeared to some adrift and old-fashioned. Profits were down. Its mutual funds’ star stock pickers were losing clients. A firm that once stood as the world’s biggest money manager had slipped behind BlackRock Inc. and Vanguard Group. Some inside the family-controlled firm worried Johnson wasn’t bold enough to lead it.