One early match in a Google Books search for "cut me some slack" occurs in a glossary purportedly from Jack Sweetman, The U.S. Naval Academy, an Illustrated History (1979) [snippet]:
Slack — reduced discipline; see Cut (Me) Some Slack.
Unfortunately, we can't see "Cut (Me) Some Slack" because the book refuses to show a snippet view of that entry. And further research leads me to suspect that the snippet shown is actually from a mid-1990s edition of Ross MacKenzie, Brief Points, which (unlike Sweetman's book) does have a glossary of Naval Academy terms.
In any event, the sense of cut in this instance appears to be meaning 15.a. in J. E. Lighter, Random House Historical Dictionary of American Slang (1994):
15.a. to favor with; to give. [Example:] 1972–76 Durden No Bugles 8: Aw, Sarge, cut us a break.
So "cut me some slack" = "favor me with reduced discipline" in 1970s (or 1990s) Middie-speak.
Robert Chapman & Barbara Kipfer, Dictionary of American Slang, third edition (1995) says "cut ... a break" was in use "by 1970s" and "cut ... some slack" was "1980s." That book interprets "cut [someone] some slack" as meaning "To stop pressuring or importuning someone; let someone be."
I haven't found any reference work that offers an explanation of what "cut some slack" might have meant originally, with actual slack and actual cutting in play. It is common for people working together on a rope to urge one another to "leave some slack" in the rope when possible, but I haven't come across any formal confirmation of this usage in connection with "cut ... some slack."
The Random House slang dictionary has only examples of "cut [someone] a break," not "cut [someone] some slack," but the meaning of the two phrases seems to be extremely similar.
Early confirmed Google Books matches
Another oldish Google Books match is from Revista Chicano-Riqueña, volume 7 (1979):
TERESA: I'm going to miss you, you big tonto. MANOLO: Who's tonto? TERESA: You are. MANOLO: Cut me some slack, mujer. TERESA: Tonto.
MANOLO: Chale. I don't care about that. DOMINGO: But there are doctors and clinics... MANOLO: ¡Ya! I said no! DOMINGO: Then why don't you go along with el Louie? MANOLO: Cut me some slack will you? DOMINGO: You're scared aren't you? I can see you are scared.
Earlier still is this extract from Wyoming Highway Department Public Information Office, The Roadrunner (1976) [combined snippets]:
...the officer cut him some slack. Then he went on to say how poorly he was treated and how tight the handcuffs were and how they hurt his wrists. He commented that he was peaceful and didn't need to be cuffed. He went on about how he had how he had to take the time to go to Cheyenne to appear in court and the money it cost him, etc.
By this time, I'm sure you can see why my heart was breaking in two for the guy. I let him have his say, and then proceeded to set the record straight.
And from Flying, volume 98 (1976) [combined snippets]:
Someone else had once taught me that good pilots don't accept a descent from ATC unless they are ready for it and know where they are. There are a lot of hard mountains in that area, fellas, so when we ask you to assure us of terrain clearance—we know it's still our responsibility to know that in spite of that nasty TWA mess—but cut us some slack, huh?
Neither of the 1976 magazine dates is definite, but both are circumstantially plausible. Several earlier would-be matches are clearly incorrectly dated in the Google Books database.
Early confirmed newspaper database matches
An Elephind newspaper database search finds five matches involving "cut [someone] some slack" from the 1970s—one from 1978, one from 1977, two from 1975, and one from 1972.
From "Reader Supports Driver; Backs Humorus Comment," in the [Kent, Ohio] Daily Kent Stater (May 31, 1978):
So, the next time you use the convenience of riding a campus bus, before you get on, cut the driver some slack. Maybe he’s more human than you think. (Anyway, if you’re at all good-looking, maybe he’s checking you out, too!)
From John Marchese, "Marijuana Use in Dormitories," in the [Denton, Texas] North Texas Daily (February 18, 1977):
"The first time I catch someone, I cut him some slack. If I catch him again then I'll have to take him to the director of housing."
Other dormitories are not so lenient, but most RAs "cut some slack" on the first offense.
From "Early Out During Xmas," in the [Temple, Texas] Fort Hood Sentinel (December 4, 1975):
Reserve component personnel undergoing active training and officers involuntarily separating on specified dates as directed by DA [Department of the Army] will separate regardless of the policy.
The policy, however, cuts some slack for local commands. Officers who are involved in priority operations may be retained until their normal ETS [expiration of term of service] date if it is necessary.
From "Misclassifieds," in the [Houston, Texas] Rice Thresher (April 24, 1975):
Rob—cut me some slack!—Graz.
Leiter—Cut me some slack.— Rob.
And from "Smoking Dopes," in the [State College, Pennsylvania] Daily Collegian (November 9, 1972):
I suppose the reader may be asking himself, "doesn't this guy appreciate a cheap drunk; a chance to get up on quality hash for free; an opportunity to groove on Isaac [Hates], while getting vicariously stoned? "At the risk of sounding like an ingrate" —Hell no, I keenly resent it!
I see three possible means of rectifying this respiratory rape:
- Students attending concerts can cut the non-heads in attendance some slack by leaving their "J's" at home instead of ruthlessly bludgeoning these fellow students into submission with them.
These five instances from 1972–1978 (all of which come from U.S. university or military base sources) seem to use the expression "cut [someone] a break" in the still-current sense of "go easy on [someone]," "be lenient with [someone]," or "give [someone] someone some leeway."
And finally, as Jacob Lyles points out in a comment beneath this answer, a list of new expressions in Idioma, volumes 5–6 (1968), page 58, includes this entry (and translation):
Hey, man, cut me some slack. [Translation:] Hey, buddy, take it easy.
A more accurate translation of this wording might be "Hey, buddy, loosen up on me."