peeve (v.)

late 14c., peyvesshe "perverse, capricious, silly," a word of uncertain origin; probably modeled on Latin perversus "reversed, perverse," past participle of pervertere "to turn about" (see pervert (v.)). Original sense was obsolete from 17c. The meaning "cross, fretful. ill-tempered" is recorded from 1520s. Related: Peevishly; peevishness.

"thing that provokes one most," by 1917, from pet (n.1) in the adjectival sense "especially cherished thing" (1826), here in jocular or ironic use with peeve (n.) and perhaps a suggestion of pet (n.2).