You may want to take this one to /r/Etymology, but meanwhile, a quick stop in etymonline.com reveals this info:

Rhine: principal river in western Germany, from German Rhein, from Middle High German Rin, ultimately from Gaulish Renos, literally "that which flows," from PIE root *reie- "to move, flow, run" (cognates: 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"). The spelling with -h- (also in Latin Rhenus; French Rhin) is from influence of the Greek form of the name, Rhenos.

and

Rhone: river in southeastern France, from a pre-Indo-European element *rod- meaning "to flow."