I always knew it as a shortened version of "copper," but the etymology had nothing to do with the badge (badges were usually made of tin, I thought). The term started in Britain, not NYC - and I know they still use the term "copper" in England, because of some DVDs of modern English crime dramas I have.
I looked it up.
From Wikipedia:
Cop, Coppa or Copper
The term copper was the original, unshortened word, originally used in Britain to mean "someone who captures". In British English, the term cop is recorded (Shorter Oxford Dictionary) in the sense of 'to capture' from 1704, derived from the Latin capere via the Old French caper. There is a common but mistaken belief that it refers to the police uniform's buttons or badge being made of copper.
From
http://orvillejenkins.com/words/cops.html:
Dr. Orville Boyd Jenkins
Question:
Why are Police Called Cops?
Answer:
Some have proposed that the reason police officers are called "cops" is because of the New York force's copper badges. The source of this word is much earlier. Several sources say the usages originated in Northern England.
The word "cop" is an old Anglo-Saxon verb for catch, grab or capture, deriving from a noun "cop" dating back at least to the 1100s. Some sources say this word related to the Dutch word kapen, with a similar meaning. The earliest written documentation of the form "cop" as a verb in English dates to 1704.
A new noun form developed form this verb, giving us "copper." This form "copper" thus was the noun for "one who cops." Some sources document the use of the verb "cop" used with the meaning "arrest" in 1844, and suggests this was the source of the specific use of "cop" to refer to a law officer.
The term "copper" was originally used in England as a slang word for a police officer. This term was used, however, as a term of abuse by criminals, especially petty street criminals. It was considered highly derogatory by the police themselves. It was made illegal in England to use this term for a police officer, because it was so derogatory.
The term became associated with the metal copper after that law was passed, when those hoods who wanted to abuse an officer would carry a small piece of copper metal in the palm of their hand, then flash it at the policeman. This usage of "copper" got shortened to "cop." The first documented use of the term copper for a police officer is in 1846. The Oxford English Dictionary documents this usage. Most sources report that the first documented use of the short form cop as applied to a police officer is in 1859. Snopes finds that the term "copper" was used in London for police by 1846.
Even when I was a child in the US in the 1950s, though not illegal, it was still considered impolite to call a policeman a cop. We were taught it was only the bad boys who feared being caught by the police that called policemen cops. (It is only in comparatively recent history that police themselves have accepted and used the word for themselves.)
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So, yeah, that all makes sense to me. When you "cop" something, you capture it, in a sense. When I was a teenager in the '70s, we used to say "cop a squat" to mean "grab a seat." A copper is one who cops.
