Maybe someone else can help out with the origin of 'cut', but "give me some slack" is an expression that originated during the docking of a ship. One would alternately tension the line in your hands and then release. The call would be to "give me some slack" when it was your turn to "haul".

"slack" as a verb generally meant to release tension, i.e. "slack the laniard of our main-stay".

"slack" is otherwise considered a bad thing. To be "slack in stays" means your ship was slow in going about or applied to a lazy man. A "slack helm" means your ship is 'too much by the stern', which carries the helm a-lee.

Source: The Sailor's Word Book, Adm. W.H. Smyth (1867)

"Slack" is a rather old English word. The Oxford English Dictionary indicates its use as early as 897 to mean "idle, lazy." It still has this same meaning as a verb: slack off, slacker, etc. Chaucer used it by 1386 to mean "loose, not taught," a pretty straightforward transition. This is the meaning the phrase comes from.

The noun "slack" refers to the slack part of something. Since sailing has a lot of ropes, its been in common use for years in that field. D. Steel's 1794 Elements of Rigging and Seamanship indicated that "Topsails are allowed 3 inches slack in every foot in the cloth." Slack rope allows for both sudden tension on ropes, such as a blast of wind on sails, and freer movement, as when tying a knot to join two ropes, knitting a sweater, or climbing a rock face with a belay. I hope you can see the metaphorical connection there.

With Google's Ngram, we can see that the specific phrase "give him some slack" was used as early as 1900. "Cut him some slack" doesn't appear until the 1960s, put the phrase "to cut slack" is from as early as 1895.