I understand what it is, like an item proc that delivers an effect on hit for example. But what does PROC actually mean? Does it stand for something or is it short for another word?
Apparently, there are several definitions available.
1) It stands for "Programmed Random OCcurence". This term pretty accurately describes the event itself, both its randomness and the fact there is some special effect.
2) Short for Procedure, or Special Procedure. Coming far back from the old text MUDs (Multi-user dungeons), it stands for the special procedure, or event, that happens when the stars are right and a random number fits some required range.
Being a programmer, I think of it as derived from "Procedure" (though what I think doesn't really affect where it originally came from). As an example, say you attack an enemy unit. As your attack strikes, an on-hit event occurs and all your items and abilities execute their procedures for that event - they "proc".
Programmed Random OCcurence doesn't sit well with me, especially w.r.t. LoL - does LoL even have any random procs left? The acronym may well be etymological, but even if that's the case it no longer has anything to do with the term "proc" as it's being used today.
Since "Procedure" is a bit technical and "PROc" has lost any potential meanting, I don't think "proc" should be considered an abbreviation or acronym for anything - it's just jargon.
An answer by way of example:
Let's say that I have a sword which does 16 damage but is also endowed with a 50% chance to freeze on hit.
I wouldn't say: "I
proc the sword."
But if it hit I would say: "The sword
Using the term loosely in this situation I might even say: "I
proced the freeze."
It's worth mentioning in an answer that this is related to: On the meaning of 'proc'
Which I think probably has the best exposition on a definition.