'nonrhyming' is a compound formed in Modern English from the negative prefix 'non-' (from Latin 'non', meaning 'not') attached to 'rhyming' (the present participle of 'rhyme').

'rhyme' came into Middle English as Old French 'rime' (from Medieval Latin/rhythm-related forms) and later became the modern English word 'rhyme'; the prefix 'non-' (from Latin 'non') was productively combined with participles and adjectives in Modern English to form compounds like 'non-rhyming'.

Originally related words for 'rhyme' derived from Greek 'rhuthmos' (meaning 'rhythm' or 'measured motion'); over time the meaning shifted to the modern sense of correspondence of sounds at the ends of lines, so 'nonrhyming' now means 'not exhibiting that correspondence of sound'.