What's the etymology of the expression "let it slide?"

ODO's entry for slide defines "let something slide" as:

negligently allow something to deteriorate:
Papa had let the business slide after Mama’s death

Therefore, the connotation is quite literal in that you are letting something slide down. Furthermore, there is also an implicit restriction involved in your expression which would expand to something along the lines of:

Let it slide this time

In other words, you're allowing the situation to deteriorate temporarily.

The etymology of the word slide dates the first use of let (something) slide to the 14^th century:

O.E. slidan "move smoothly, glide," from P.Gmc. *slidanan (cf. O.H.G. slito, Ger. Schlitten "sledge"), from PIE root *(s)lei- "slide" (cf. Lith. slystu "to glide, slide," O.C.S. sledu "track," Gk. olisthos "slipperiness," olisthanein "to slip," M.Ir. sloet "slide"). Phrase to let (something) slide is recorded from late 14c.