What exactly is a "ring-a-ding girl"?

An AmE expression from the ‘60s:

ring-a-ding adj. (the image of celebratory bell-ringing)

perfect, ideal.

• 1960 [US] ‘Lord Buckley’ Hiparama of the Classics 22: He’s havin’ a ball of all balls, happy as a ring a ding bird in a ding dong tree.

• 1962 [US] Mad mag. Dec. 24: Frank Sinatra here, with a ring-a-ding home movie.

• 1995 [US] A. Heckerling Clueless [film script] So you didn’t want to make a night of it with the ring-a-ding kid?

(GDoS)

The earliest version of The American Slang Dictionary (which first appeared in 1960) include an entry for ring-a-ding is Robert Chapman, New Dictionary of American Slang (1986):

ring-a-ding-ding or ring-a-ding 1 n Glamor and show; spectacular impressiveness; = RAZZLE-DAZZLE: an aura of breathless showbiz ring-a-ding-ding—Albert Goldman 2 adj: Our new stack addition is a huge brick building full of metal, a ring-a-doing book box—American Libraries

Tom Dalzell & Terry Victor, The New Partridge Dictionary of Slang and Unconventional English (2006) ha this entry:

ring-a-ding noun an excellent example of something US [Cited example:] In the patois of the Rat Pack, a ring a ding of a scene.—Sidney Bernard, This Way to the Apocalypse, p. 153, 1965

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'Ring-a-ding' as a bell sound in Australia, 1869–1948

In Australia, evidence of "ring-a-ding" as an onomatopoetic word for chiming bells goes back to at least 1869, in the verses of a sailors' song. From "A Fog on the Banks of Newfoundland" in the [Wollongong, New South Wales] Illawarra Mercury (December 28, 1869):

Upon Newfoundland's dreary coast, / Midst fog, and ice, and storm, / Three miserable days we past, / Fired guns and blew fog-horn; / Phoot, phoot, and bang, and ring a ding / Went horn, and gun, and bell, / Masses of ice which round us clung, / Hove with the heavy swell.

We were on the "Great Bank," shoals of fish in every direction; the demand for hooks and lines exceeded the supply, but our "take" was enormous, halibut, cod, and ling, weighing from five to a hundred and fifty pounds; when tired of catching, we turned to and salted enough for a six months stock. The following morning found us enveloped in a thick fog; ... Bring up the ""fog horn," and keep the bell going; a fog horn is a long trumpet-shaped instrument which gives a loud and particularly disagreeable sound; phoot, poot, phoot, went the horn; ring-a-ding, ring a-ding, the ship's bells, as loud as they could ring, and a gun was fired every half hour; ...

Over several years, starting in 1890, various Australian newspapers reprint a poem titled "School-Room Jingles" that contains multiple instances of "ring-a-ding-ding" and ring-a-ding-dong." And other occurrences, including a ones involving racehorse named "Ring-a-Ding," pop up in Australian papers in subsequent decades.

"Ring a ding" appears occasionally in U.S. newspapers in the bell-sounding sense as early as 1904. From "Just Before Breakfast," in the Richmond [Indiana] Palladium (October 20, 1904):

Over the phone—No, —— please. What number? I said ——, didn't I? No response for a time. Ring a ding ring! Hello, give me ——, please. Did yon say ——? I guess you heard what I said and if I don't get it I'll know the reason why.

However, the usage was quite uncommon (and unhyphenated) in the United States until the late 1950s.

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'Ring-a-ding' as U.S. slang, from 1958 through 1963

The earliest U.S. newspaper instance of "ring-a-ding" that an Elephind search turns up is much later: in 1958, in a discussion of television slang. From William Morris, "Words, Wit & Wisdom: TV Talk," in the San Bernardino [California] Sun (June 2, 1958):

One technique used by Steve Allen to retain freshness in a performance is to advise his fellow performers to hold the punch lines of a skit during rehearsal, speaking them only on the show itself. Allen's phrase is "ring-a-ding it" — meaning hold the gag for air time.

A year later, this item, appears in "To Appear on Tyler TV Sunday," in the Rusk [Texas] Cherokeean (August 27, 1959):

Marcy Goff will appear on the Town and Country Variety Show, KLTV, Tyler, Sunday afternoon, August 30th from 3 to 4.

Miss Goff, the talented daughter of Mr. and rs. Lester Goff, has made several appearances on KLTV, and at Bergfeld Park in Tyler this summer. She specializes in tap, and her number will be Tommy Sands' "Ring-a-Ding."

Sands released his song "Ring-a-Ding-a-Ding" in 1957. In the song, "ring-a-ding-a-ding" is what the singer's heart goes, thanks to a kiss from his girlfriend. According to the person who posted a recording of the song on YouTube, it was a followup to his debut hit "Teenage Crush." One can only imagine the tap-dancing possibilities of this number.

Use of "ring-a-ding" as an adjective evidently received a major jolt from an ad campaign for Ripple Wine ("the new wine for lively people — made by Gallo"). Starting early in 1960, ads like this one from the San Bernardino [California] Sun (January 22, 1960) began appearing in newspapers:

GALLO RIPPLE WINE A new wine for every one to taste — Its "Ring-a-ding" flavor will refresh you any time. Enjoy both red and white Ripple. .......TENTH 25¢ CASE OF 24 TENTHS 5.40

Noncommercial instances followed. From Mike Connolly, "R-Week for Weaver," in the [Palm Springs, California] Desert Sun (April 21, 1960):

Tose internecine tussles during the tryout tour of Maureen O’Hara’s new musical, "Christine," are now a thing of the past but Wow. what a ring-a-ding cat-and-dog duel preceded the peace-pipe puffing! It started when the producer fired director Jerome Chodorov as director and Pearl Buck (she adapted the play from her own book).

And from Mike Connolly (again), "Still Seek Formula," in the [Palm Springs, California] Desert Sun (April 25, 1960):

The Los Angeles Chamber of Commerce and the fair city's Fire Department are having themselves a regular little old ring-a-ding feud these days. I’m sure you are all more than dimly aware, thanks to every-hour-on-the-hour puffs of publicity smoke from our heap big Made-in-Albuquerque, dyed-in-the-wool genuine Indian smoke blankets, that a certain political party or two will stage its (or their) convention(s) 'way out here where the West begins sometime this summer (and if I don't get to the point of this story it'll be suddenly LAST summer).

From "'Bells Are Ringing' at Central Theater," in the [Pearl River, New York] Orangetown Telegram (August 11, 1960):

The ring-a-ding musical stars the talented Judy Holliday recreating her stage success. Co-starred are Dean Martin, Fred Clark, and Eddie Foy, Jr.

From "Brook Benton, The Drifters Play Concert in Muncie Sat., Nov. 19," in the Indianapolis [Indiana] Recorder (November 19, 1960):

[DIDJA KNOW] THAT out in California—among the many commercials used against Kennedy was this cartoon which presents a "Mr. Beatnik," who says, "Well, man, if I can drag myself to the polls I'll vote for that new frontier kid. Why man, I dig that cradle-to-grave security bit ... Nothing to do but stay in your pad and listen to that ring-a-ding jazz all day while Uncle Whiskers pays for the rent and the wine[.] Cool, man, it's the craziest." When a character titled "Emcee" breaks in with a query about the withholding tax rising, "Mr. Beatnik" answers: "Withholding tax? Why man, you have to work to pay withholding tax, and only squares work."

And from an advertisement for Polka Dotta in the Palos Verdes Peninsula News (December 15, 1960):

Ten nights before Christmas and all through the house / We can hear you men shouting, where can I find a blouse, / or some shoes / or a skirt, / or any old thing / For the girl of my dreams — who'so ring-a-ding ding. / Gentlemen ... Unite! You have nothing to fear, / Polka Dotta will serve you AND Free Pretzels and Beer / The store will be open Wedn[e]sday Night — Just for you! / Come buy all your gifts and have them gift wrapped too!

The Twilight Zone episode "Ring-a-Ding Girl" seems to have aired on the night of December 27, 1963, at which point "ring-a-ding" was evidently in widespread use in U.S. slang as an adjective signifying (as Chapman suggests) "spectacular and impressive." It seems quite significant that the phrase "ring-a-ding girl" appears nine times in Elephind search results between 1963 and 1980—but only in connection with this episode of The Twilight Zone.

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Similar-looking terms with distinctly different meanings in U.S. slang

Several terms that have had some currency in U.S. slang are strongly suggestive of "ring-a-ding" but have different senses that aren't especially close to "excellent," "outstanding," "razzle-dazzle," "spectacularly impressive," or the like. For reference, I list the entries for these other terms as they appear in Robert Chapman & Barbara Kipfer, Dictionary of American Slang, third edition (1995):

ding-a-ling 1 n by 1930s An eccentric person = NUT, SCREWBALL [citation omitted] 2 adj by 1930s [citation omitted] {fr the notion that such a person hears bells ringing in the head}

ding-dong n by 1970s A stupid person; idiot; =DINGBAT

ding-dong 1 adj by 1870 Vigorous and spirited; =KNOCK-DOWN-DRAG-OUT | Used adverbially "with a will" by 1672 [[citation omitted] 2 n by 1920s An eccentric person; =DING-A-LING, NUT 3 by 1940s The penis; =DONG [citation omitted]

ring-dang-do n by 1970s A complicated process, scene, affair, etc.; rigmarole; =RIGMATICK [citation omitted]

ring-ding n by 1970s A stupid person; =DING-A-LING [citation omitted]

Although there is a lot of overlap in meaning among these terms, the only element in them that seems clearly similar to any of the concrete instances of usage of "ring-a-ding" from the early 1960s cited earlier in this answer is the "vigorous and spirited; KNOCK=DOWN-DRAG=OUT" sense of "ding-dong," which seems to be the core sense of "ring-a-ding" as used by Mike Connolly (twice) in articles published in the Palm Springs Desert Sun in April 1960. Connolly's sense of the term doesn't seem to have become widespread in U.S. usage, however, and by late 1963 the term appears in U.S. newspaper articles very nearly exclusively in the "excellent, spectacular, impressive" sense.

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Conclusions

In U.S. slang usage "ring-a-ding" gives every indication of having arisen a show-biz/press-agent term for something big, boffo, and bankable. The verb sense of "ring-a-ding" as "save it for showtime" seems to have originated with Steve Allen in his Tonight Show years and to have vanished thereafter. But it might have been picked up by Allen fans in adjective form to signify "razzle-dazzle."

By late 1963, when the word appeared in the episode title "The Ring-a-Ding Girl," the sense of the term as a modifier seems to have been well established as "excellent, stupendous, superlative." The longer expression "ring-a-ding girl" does not seem to have existed as a separate idiomatic set phrase in U.S. slang, so the "ring-a-ding girl" in the Twilight Zone episode was presumably so-called because of her unusual abilities or powers.

A dictionary definition

ring-a-ding in American English (ˈrɪŋəˌdɪŋ ) US, Slang ADJECTIVE

1. wildly exciting NOUN
2. wild excitement; razzle-dazzle
3. a wildly exciting person or thing Webster’s New World College Dictionary, 4th Edition. Copyright © 2010 by Houghton Mifflin Harcourt. All rights reserved.

(Collins Dictionary)