"a conceited, narrow-minded pragmatical person; a dull, precise person; one who cultivates or affects propriety and offends or bores others," 1753, originally in reference to theological scruples (1704), a word of unknown origin.
It could be related to earlier appearances of the same word meaning "a dandy, coxcomb, fop" (1670s), "thief" (c. 1600; in forms prigger, prigman recorded from 1560s). Century Dictionary speculates the modern word is "perhaps a later application (of the "thief" sense) in the general sense, among "the profession," of 'a smart fellow.' " Also compare thieves' cant prig "a tinker" (1560s). In Middle English a prig was a kind of small nail used in roofing or tiling (14c.), perhaps from prick.
A p[rig] is wise beyond his years in all the things that do not matter. A p. cracks nuts with a steam hammer: that is, calls in the first principles of morality to decide whether he may, or must, do something of as little importance as drinking a glass of beer. On the whole, one may, perhaps, say that all his different characteristics come from the combination, in varying proportions, of three things—the desire to do his duty, the belief that he knows better than other people, & blindness to the difference in value between different things. [quoted in Fowler, 1926, who writes that it can be found in "an anonymous volume of essays"]