Early in Marilyn Maye’s latest show at 54 Below, she introduces “Here’s that Rainy Day” with the well-known fact that this classic torch song by Johnny Burke and Jimmy Van Heusen was the all-time favorite of a legendary late-night talk show host, Johnny Carson. 

Even though it’s been roughly 50 years since she sang this song for Carson on “The Tonight Show,” she still expresses surprise that the most celebrated comedian in America would name such a melancholy, even heartbreaking song — with its famous opening line about saving “leftover dreams” — as his fave. 

Ms. Maye, who just returned from participating in a celebration of Carson’s centennial at his alma mater, the University of Nebraska, told a story about how the host would visit her immediately before the taping of each show. “He would pop into the makeup room and he would say, ‘Marilyn, are you going to sing “Rainy Day?” And I would say, ‘Well Johnny, I’ve already done it three times on the show.’ He would say, ‘That’s fine. You can do it as many times as you want.’”

Some singers will come out and talk about their hit records, but for Marilyn Maye — who is tragically under-recorded for a singer of her stature — the no. 1 metric of her success is the support she earned from the three major hosts of “The Tonight Show,” Steve Allen, Jack Paar, and especially Carson. She reports having appeared on the program 76 times, and she shows a circa 1976 clip of Carson introducing her in which he mentions that this was already her 60th appearance.

At another point during her show, Ms. Maye apologizes for not knowing the words to a certain song. It’s not even one that’s on her set list, but it gives her the chance to crack a joke: “You’ll forgive me because I’m 79.” She doesn’t have to wait long for a big laugh, because everyone in the room knows that those digits are reversed: Ms. Maye is actually 97.

The surprising thing is that she’s been working not only more frequently but harder than ever in recent years. Most of us in the house have seen her so many times that we could practically sing her familiar medleys along with her. But in the last few seasons, she’s made a point to insert more and more unfamiliar material into the act, songs that are not exactly new but that she hasn’t sung in a while. As you can imagine, at 97 she has a lot to choose from.

The current show is very much in keeping with the theme of the Carson centennial. There are few traditional songbook standards: The most notable are “Rainy Day” mashed up with “Stormy Weather” and her sweet, intimate closer, “That’s All.” Rather, most everything she sings this week is from the 1970s, many of them being songs she actually sang on “The Tonight Show” itself. There was quite a lot that even us regulars hadn’t heard, in particular Jim Weatherly’s “The Need to Be” (1974), which she learned from Gladys Knight, and Olivia Newton John’s hit “Let Me Be There.” 

There was a “new” medley — or at least one that I had never heard before — of “time” songs, which centered around Carole King’s “It’s Gonna Take Some Time.” Ms. Maye has a gift of taking tunes you wouldn’t know have anything in common and making them work together, like the Russian gypsy-styled hit “Those Were the Days” and the disco anthem “I Will Survive,” which she somehow transforms from a hoary cliche into an ode of genuine emotional resonance.  

She can even take two songs that are completely inapposite — “By Myself,” which is about being resigned to that condition, and “Being Alive,” which is about connection at all costs — and forge them into a coherent statement.

For the ending, she gave us some more vintage songs of the season, “Autumn In New York” followed by “Autumn Leaves” and “When October Goes,” and then “That’s All.” Lest this sound like the show was overly melancholy in an autumnal sort of way, there was humor galore, mostly in the form of her spontaneous quips, which she delivers with perfect timing. Imagine if George Burns and Gracie Allen were both the same person.  

You couldn’t help but feel that the University of Nebraska had to have been overjoyed to find a veteran of “The Tonight Show” who is so close to Carson’s own age and yet so incredibly spry. All of which was a bittersweet reminder that the tradition Carson perfected is very much under fire — not merely because so many current  comics and hosts are being threatened for speaking truth to power but because the late-night talk show format itself is an increasingly unprofitable medium and may not be around much longer.  

In fact, Marilyn Maye herself will likely outlive it. Thank God we have her to remind us to save those leftover dreams.