It was a dark and stormy night at Paramount Pictures when Gloria Calderon Kellett and Mike Royce sat down to write the revival of their hit 1975 sitcom, One Day at a Time. But Mr. Royce (who directed the film version) looked to the chaotic halls, and he saw hope.

“‘There’s opportunity to make this fun again,’” Mr. Royce recalled telling Ms. Calderon Kellett, co-writer of the original series. The show had, after all, been nominated for 10 Emmy Awards and won two. In its original run, One Day at a Time, about a middle-aged single mom trying to juggle a career and raising two kids, ran for eight seasons.

Some outlets seemed certain the show would fail. “The Weather Man is trying his hand at a revival: AMTeVe cancels weekly comedy. Come back in 2020,” read a headline on The Wrap, an entertainment media company.

“We had four or five times that many people phone me,” said Carol Mendelsohn, the creator and producer of The Mentalist. “‘I’m so depressed. I don’t know if I can do this again.’ I heard that from so many people.”

Ms. Mendelsohn and Ms. Calderon Kellett, both mothers, worried about the show’s target audience — but they were having fun in the writer’s room. They were joking around with each other and casting the show’s updated family and friends. As the crowd started to fill up, the anxiety started to subside.

“I saw this huge movement of people,” Ms. Calderon Kellett said. “That’s when we knew we had a shot.”