Christopher James’s decision to do battle with one of the world’s biggest oil companies began with a family dinner in 2019. The hedge-fund manager’s school-aged sons asked him how he could consider himself an environmentalist if he invested in energy companies, and Mr. James said he struggled with his explanation.

“As I was listening to myself talk, I thought `I am really splitting hairs on this.’” One of his sons, he said, “had this look on his face where his forehead wrinkled. He didn’t buy it.”

Two years later, Mr. James convinced Wall Street’s biggest money managers to back a dramatic change in board oversight at Exxon Mobil Corp. , a watershed moment in the effort seeking more environmental, social and governance modifications at the nation’s biggest companies. He contributed nearly $250 million of his own money to support the campaign and launch a new fund called Engine No. 1 LLC. The resulting shareholder vote, tallied last month, secured board seats for three of his handpicked candidates.

Mr. James, 51 years old, was an unlikely catalyst for change at an energy giant. After making a name for himself during the dot-com boom and bust, he operated an Illinois coal mine and built storage facilities used by the oil-and-gas companies. Away from work, however, he supported conservationist causes. The Exxon campaign offered a chance to align his personal values with an investment thesis—that the giants of the oil industry would drop in value unless they embraced a transition to renewable energy.

Mr. James’s hedge fund called on Exxon to slash expenses on projects that might lose money when oil and gas prices are low, realign management incentives and develop a plan to invest in renewable energy. Pictured here is an Exxon refinery in Baton Rouge, La.

He also proved he could make his case for change without any of the usual fiery tactics activists employ to lift share prices in a hurry. Engine No. 1’s list of demands, instead, read more like a corporate sustainability report. In its December note to Exxon’s board, Mr. James’s fund called on the company to slash expenses on projects that might lose money when oil and gas prices are low, realign management incentives and develop a plan to invest in renewable energy.

“If we’re right on getting Exxon to mitigate these impacts, the stock should go up,” Mr. James said in an interview. “And maybe Exxon does have a future.”

So far, the fund has made money on that investment, he said. It is now seeking to raise outside capital and launch an exchange-traded fund. Exxon’s shares, which Engine No. 1 bought at around $36, have climbed above $62. The stock closed Friday at $62.17. A rally in oil prices has helped boost energy companies’ shares in recent weeks.