rhyme

First recorded in 1250–1300; Middle English rime, from Old French, derivative of rimer “to rhyme,” from unattested Gallo-Romance rimāre “to put in a row,” ultimately derived from Old High German rīm “series, row”; probably not connected with Latin rhythmus “rhythm,” although current spelling (from about 1600) is apparently by association with this word

The spelling and etymology of the noun rhyme fall between two stools. Its Middle English forms rym (in The Canterbury Tales, from around 1387), ryym (in Wycliffe’s Bible ), and rime derive from Anglo-French, Old French, and Middle French rime, ryme. Note the absence of h in all these spellings.
The source of the French rime is from an unrecorded Gallo-Romance verb rimāre “to set in a row,” a derivative of the Germanic noun rīm “number, series,” and possibly developing the senses “series of rhymed syllables” and “rhymed verse.”
The English spelling rhyme dates from around 1600 and shows the influence of the unrelated Latin rhetorical term rhythmus “a patterned sequence of sounds; measured flow of words or phrases in prose,” a borrowing from Greek rhythmós, which has the same meanings.