This question is surprisingly difficult to answer using Google, mostly because there was a 2004 movie by the name Little Black Book. The question is about the actual term, since the concept itself obviously predates Cheers. Are there earlier examples of a TV or movie character mentioning their little black book?
See the Odd Couple in 1970 for an earlier mention
Roger Watters of Pink Floyd had a little black book with his poems in it back in 1979 on the Wall album. Little black books with supposed names and phone numbers of available women was a common term dating back at least into the 20’s.
I believe that, in the 1970s, in Happy Days, Fonzie, who famously dated a lot of women, had a “little black book.” Without going back and watching episodes, I’m not sure that I can prove that the phrase itself was used on the show, the IMDB entry for the Season 3 (1975) episode, “A Date With Fonzie,” has the capsule summary: “The Fonz picks two of his favorite chicks out of his not-so-little black book to help Richie out of a dating dry spell.” (The two women referenced were the first appearances of the Laverne and Shirley characters. 
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Not that it predates some of the other mentions, but Tony on Who’s The Boss and I believe Jessie on Full House had them.
From wiki:
The 1953 film version of Kiss Me, Kate features a musical scene in which Howard Keel’s character laments the loss of the social life he enjoyed before marriage, naming numerous female romantic encounters while perusing a miniature black book, which has given rise to the trope of a little black book referring to a list of past or potential sexual partners.
I’d be rather certain the term little black book and it’s meaning goes back centuries even before the time of Don Juan/Don Giovanni.
It’s use is archaic today, what’s the 2022 equivalent? A you tube channel lol
Oh Fonzie! Why do you hate Richie so much?
From a comic book I had in the 60’s:
Veronica: my mother said not to trust a man with a little black book.
Archie: what color does your mother prefer?
Originally known as a “minute parchment codex”. Although, being a list of phone numbers, its true usefulness had to wait for the invention of the telephone, as yelling a string of random numbers out your window rarely resulted in getting laid.
I recall an episode of Happy Days where gangsters tried to kill Fonzie by blowing up his garage. He survived the explosion because he was shielded by his little black book–which was actually several steel filing cabinets.
Wow, that really was a ridiculous show, wasn’t it? And yet, I remember being a regular viewer and discussing it in school the next day.
There’s this hit song from 1962. I’m certain it was a well known phrase then.
Me, too. That was a two-parter. The explosion happened at the end of part one. I remember impassioned discussions with my friends all the following week. We were genuinely worried that they might actually kill off Fonzie!
And, the lyrics make it pretty clear that the “little black book” is exactly what the OP is referring to. The first verse:
The Wikipedia article below dates it to the 1953 film version of the musical Kiss Me, Kate. This is for the usage of it as a small book where people wrote down telephone numbers particularly of dating partners. It couldn’t really be much before that because direct dialing people (not using an operator to connect) was new then.
Of course, there was also the “little red book” which served the same purpose.
The phone numbers are modern but the list of conquests, courtesans and ladies is ancient.
Definitely. I recall an episode of Voyage to the Bottom of the Sea where, in an emergency situation where a couple of officers (on dry land) need to start a fire, one of them refuses to give up the paper in his “little black book”. I was a kid then, and had never heard the phrase before, and had no idea what it meant.
Yeah. I remember that. Laverne, Shirley and Carmin attended Fonzie’s (fake) funeral.
Obviously. But this the question is about the specific phrase Little Black Book which is a thing that specifically has telephone numbers.