Where does this word come from? Are these ancillary benefits called this because they perk you up from your otherwise dreary worklife? Or are they somehow related to the reinvigorating effect of a cup of coffee (does anyone actually refer to "perked coffee" anymore?).
No, in fact "perk" is an abbreviation of the much more highfalutin word "perquisite", derived from Latin perquisitum (a thing acquired or gained, an acquisition, profit), from the same root as "acquisition". This word has had the meaning of "profits, benefits, etc. in addition to the regular revenue" since the 16th century. The slang abbreviation "perk" started to be used in the mid-19th century. I would say that "perk" has now become the neutral-register term for this concept, and that "perquisite" has become quite formal.
There is some early evidence of this being spelled "perq", because of its derivation, but current dictionaries list only "perk". Do not spell it "perq" unless you want people to think you are insufferably pretentious.
In reference to coffee, as in
‘I thought as I spotted the coffee perk, I need lethal doses of almost dangerously potent coffee.’or as the verb
‘A pot of coffee already sat perking, filling the room with its rich aroma.’"perk" is obviously derived from "percolate" and "percolator". And yet no one suggests it should be spelled "perc", hmmmm. These words come from Latin percōlāre to filter, strain, trickle through < per- through + cōlāre filter.
Perk up, meaning "become or make more cheerful, lively, or interesting." is unrelated, but of uncertain origin. It may be related to "perch".