"small stream or brook," 1580s, perhaps from Italian rivoletto, diminutive of rivolo, itself a diminutive of rivo "brook," from Latin rivus "stream, brook" (from PIE *reiwos "that which flows," from root *rei- "to run, flow"). For ending, see -let.
It is the hypothetical source of/evidence for its existence is provided by: Sanskrit rinati "causes to flow," ritih "stream, course;" Latin rivus "stream;" Old Church Slavonic reka "river;" Middle Irish rian "river, way;" Gothic rinnan "run, flow," rinno "brook;" Middle Low German ride "brook;" Old English riþ "stream;" Old English rinnan, Old Norse rinna "to run," Dutch ril "running stream."
diminutive noun-forming element, Middle English, from Old French -elet, which often is a double-diminutive. It consists of Old French diminutive -et, -ette (see -et) added to nouns in -el, which in many cases represents Latin diminutive -ellus; see -el (2)). "The formation did not become common until the 18th c." [OED].
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<a href="https://www.etymonline.com/word/rivulet">Etymology of rivulet by etymonline</a>
Harper, D. (n.d.). Etymology of rivulet. Online Etymology Dictionary. Retrieved $(datetime), from https://www.etymonline.com/word/rivulet