'peeve' originates as a back-formation from the adjective 'peevish' in early 20th-century American English.

'peevish' was recorded earlier in English (from Middle/Early Modern English usage) meaning 'irritable' or 'cross'; speakers removed the suffix '-ish' to form the noun and verb 'peeve', which became common in modern English in the early 1900s.

Initially related to the adjective meaning 'irritable' (peevish = 'easily annoyed'), the back-formation 'peeve' came to mean specifically 'a cause of irritation' (noun) and 'to annoy' (verb).