"agreement in terminal sounds," 1560s, partially restored spelling, from Middle English ryme, rime (c. 1200) "measure, meter, rhythm," later "rhymed verse" (mid-13c.), from Old French rime (fem.), related to Old Provençal rim (masc.), earlier *ritme, from Latin rithmus, from Greek rhythmos "measured flow or movement, rhythm; proportion, symmetry; arrangement, order; form, shape, wise, manner; soul, disposition," related to rhein "to flow,"from PIE root *sreu- "to flow."
In Medieval Latin, rithmus was used for accentual, as opposed to quantitative, verse, and accentual verse usually was rhymed, hence the sense shift. Persistence of older form is due to popular association with Old English rim "number," from PIE root *re- "to reason, count." Phrase rhyme or reason "good sense" (chiefly used in the negative) is from late 15c. (see reason (n.)). Rhyme scheme is attested from 1931. Rhyme royal (1841) is a stanza of seven 10-syllable lines rhymed a-b-a-b-b-c-c.