These dozen or so phone conversations took place when the comedic heart of “Late Night” had shifted from scripted comedy to the compelling personality of its host. Mr. Letterman surrounded himself with a series of vivid personalities — like Chris Elliott’s bizarre characters — who inspired ad-libs that revealed more of him than a monologue joke ever could.

When Mr. Letterman moved to CBS in 1993 to begin “Late Show,” the role of his mother changed, shifting from an offstage voice to an on-air correspondent, reporting from her home in Indiana and at three different Winter Olympics, including one in Norway, where she interviewed Hillary Clinton.

Through these remote videos, she proved unerringly polite and far more game than her son. (It’s hard to imagine Mr. Letterman in a tight yellow suit and joining a luge team.) These cameos were perfectly in sync with one of the main tactics of Mr. Letterman’s TV career: turning ordinary people into funny performers while also recasting the talk show, ripping it out of its context and placing it on an airplane or in a person’s home or on the streets of New York.

What was striking about Ms. Mengering’s performances is how much the spotlight didn’t change her manner or style. No matter where she was or whom she was talking to, she maintained the same poise and equanimity. She didn’t mock show business, as her son did, but she appeared immune to its notorious seduction, and completely resisted being star-struck or remotely theatrical.

Mr. Letterman’s relationship with his mother was, of course, more complex than simply devoted parent and loving son. In interviews, he called her “the least demonstrative person” in the world, and he would tell a story about the first time his mother saw him perform live, when it was clear her muted response made an impact. After telling this anecdote to Mr. Stern on “Late Show” in 2013, his famously blunt listener flummoxed him with a joke: “You have mother issues, obviously, everyone knows that.”