Soon after J.J. Abrams agreed to direct the first new “Star Wars” movie in a generation, Robert Iger took him to dinner.

“I said to him, ‘You need to look at this as a $4 billion movie,’ that’s how I’m looking at it,” the Walt Disney Co. chief executive recalled in an interview, referring to the price his media conglomerate paid to buy Lucasfilm LLC in 2012.

With...