Perk has three quite separate meanings and origins, and is a microcosm of English etymology.
Its oldest use (by 1400) is as a verb, meaning to smarten (up), carry yourself in a smart or jaunty manner, or enliven. Its origin is more doubtful, though: Chambers’ Dictionary of Etymology and the OED think it may be the same word as perch, going back to Old North French and thus to the Latin pertica, rod or perch. The OED also offers a related obsolete meaning of perk as a pole, stake, bracket to support candles, or perch.
From that, we get perk up, meaning to liven or brighten up, and perky, lively, bright or even (Chambers assures me) saucy; the OED sees it as a bit more disparaging, more self-assertive or even cocky.
Perky has even achieved stardom in the animated children’s TV series Pinky and Perky (1957, revived 2008, Wikipedia). However that seems to have been a quirk of fate: its originators, Jan and Vlasta Dalibor, intended to name the anthropomorphic pigs with characteristic high-pitched voices Pinky and Porky. They changed the latter name to Perky when they hit a problem registering Porky as a character name.
Then in the nineteenth century (first recorded in 1869), the old word perquisite increasingly became abbreviated to perk, in the sense of a non-monetary benefit associated with a job or work. Thus someone working in a factory might take home an empty wooden palette on the basis that it is a perk of the job.
Perquisite is now largely disused, and derived from the Latin perquisitum, which in medieval times became an acquisition, something gained. For much of its early use in English it has referred to property gained by means other than inheritance, then casual income of the lord of a manor. By the end of the seventeenth century it was becoming used for casual fees or income in addition to the salary associated with an office or official position. In the nineteenth century it was often used to describe items which had, once they had served their purpose, been passed on to subordinates or employees to use or remove, and expected gratuities or tips.
In the twentieth century, feeling that its four letters were strong enough to bear yet another meaning, it became an abbreviation for percolator, and its original, percolate. First recorded as perc in 1934, apparently North American, it is now more commonly spelled as perk; this can be applied to the device, the coffee it produces, or verbalised to describe the action of making coffee in a percolator.
One of the early devices for producing filtered coffee, percolators (Wikipedia) anticipated the current popularity of shiny, noisy and often expensive equipment for generating ‘serious’ coffee. Their name is derived from the process of percolation, in turn from the Latin percolare, to strain through a filter.
So smarten up, a benefit in addition to pay, and filtered coffee: a lot packed into just four letters.