Debra Monk possesses a lovely voice. And a not-so-lovely one, as well.

“I have,” she explains, “the ability to sing off-key and off-tempo.” It’s a vocal talent quite apart from a lack of vocal talent. And it’s not a skill you might think would come in particularly handy for a New York stage and television actress, even one as versatile as Monk, who is far more likely to be required, at a minimum, to sing on-key and on-tempo.

Except that in “Mrs. Miller Does Her Thing,” now in a world-premiere run at Signature Theatre, sounding awful is exactly what the playwright-director, James Lapine, wants to hear from Monk. The show is about one Elva Ruby Miller, whom theatergoers of a certain age might remember as — for a brief, unshining moment — an unlikely celebrity of the 1960s. Achieving renown simply as “Mrs. Miller,” she soared to stardom on radio and television by virtue of her LPs and ghastly covers of Top-40 hits such as Petula Clark’s “Downtown,” the Beatles’ “A Hard Day’s Night” and “Monday Monday” by the Mamas and the Papas.

“She became a kind of pop sensation in spite of herself,” Lapine says, explaining that what made Miller unique was that she was in her 50s and performing excruciating versions of songs associated with a younger, hipper generation. “I don’t think it was her music, and that was the whole joke,” he says. “The question was, did she know that people were laughing at her?”

If the inquiry rings a bell, perhaps it's because a similar one was posed in "Florence Foster Jenkins," the movie from last year that earned Meryl Streep her 20th Academy Award nomination. In it, she plays a tone-deaf socialite of an earlier age, a Mrs. Miller of the 1920s and '30s, who squeaked out operatic arias so sourly that listeners' lips would pucker. Jenkins also was the subject of a 2004 off-Broadway play, "Souvenir," starring Judy Kaye, which moved the following year to Broadway, where it quickly fizzled.

This all might help to explain why Lapine, 68, didn’t have the easiest time finding a home for “Mrs. Miller Does Her Thing,” which he has been working on for the better part of a decade. Two years ago this month, though, Lapine was in Washington to receive Signature Theatre’s Sondheim Award, given to him for his distinguished collaboration as book writer and director of the Stephen Sondheim musicals “Into the Woods,” “Sunday in the Park With George” and “Passion.” That night, he mentioned “Mrs. Miller” to Signature’s artistic director, Eric Schaeffer, who agreed to produce it.

Lapine says he had long wanted to write about the music and politics of the late ’60s, and he became intrigued with Miller after being introduced to Mark Oliver Everett, a musician with the indie rock group the Eels, who had written a film treatment about her.

“I looked her up and thought, this is a great way to think about this era,” Lapine says, “because ’66 to ’68 was the middle of Vietnam and the cultural revolution, and she started recording in that period, when the music changed.”

He also long had in mind Monk, who has made a zesty career out of supporting roles: She originated the part, for example, of Sara Jane Moore in Sondheim’s 1990 musical “Assassins” and won a Tony three years later for her featured performance in a Lanford Wilson play, “Redwood Curtain.” She has been nominated for three other Tonys, and, even more impressive, the nods have come for her work in both plays and musicals. (Monk also has done television, most prominently playing the eternally worried wife of Dennis Franz’s Andy Sipowicz on “NYPD Blue.”)

Born in Ohio, Monk grew up in the Washington area and graduated from Wheaton High School. She never gave acting a moment’s consideration until after a stint as a secretary in Dupont Circle and a subsequent scholarship to Maryland’s Frostburg State College. At Frostburg, Monk was required to take a speech class and, as a result, was asked to try out for a production of an avant-garde play, Harold Pinter’s “The Birthday Party.”

“To this day, I don’t know what it’s about,” she says.

Monk did, however, have some innate belief in herself, because by her own account, she talked her way into a well-regarded graduate acting program at Southern Methodist University in Texas and, upon graduation, went to New York, where she and fellow actress Cass Morgan created a country-music musical for themselves called “Pump Boys and Dinettes.” The show would transfer in 1982 to Broadway, where it ran for nearly 600 performances.

What Monk, now in her mid-60s, hasn’t experienced much is a piece built around her. Lapine is happy to have been the supplier of that particular property.

“She really hasn’t been a headliner,” he says. “It’s a thrill to be able to give her a part that’s at the center, at this stage of her life.” (Another theater heavyweight, Boyd Gaines, is also featured in “Mrs. Miller.”)

It’s a part in which Monk gets to sing ’60s standard after ’60s standard — not just “Downtown” and “Monday Monday,” but also “These Boots Are Made for Walkin’, ” “The Ballad of the Green Berets” and “Moon River.”

Just not in the standard way.

Mrs. Miller Does Her Thing, written and directed by James Lapine. Tickets: $40-$99. Through March 26 at Signature Theatre, 4200 Campbell Ave., Arlington. Call 703-820-9771 or visit sigtheatre.org.