What is the origin of the word "copper" for referring to a police officer? [duplicate]

Solution 1:

There probably isn't a definitive reference. The Straight Dope begins its discussion of the origin of copper with this declaration:

Etymology is rarely an exact science. Words or phrases spring up, become popular, and eventually may find their way into print. The process takes time, and it's usually difficult or impossible to track backwards to discover where a particular word or phrase arose.

The article goes on to discount the speculations raised in your points #2 and #3, and flat-out says:

The notion that cop is an acronym for "Constable On Patrol" is nonsense. Similarly, the word did not arise because police uniforms in New York (or London or wherever) had copper buttons, copper badges, or anything of the sort.

It does, however, raise the alternative theory that "to cop comes from the Dutch kapen, meaning to take or to steal."

Solution 2:

It’s probably because a copper is one who cops wrong-doers. Cop in the sense of ‘capture’ or ‘catch’ is first recorded in 1704. It may be derived from the now obsolete verb cap, ‘to arrest’, from a related Old French word.