The word "Rewa" (रेवा) has been used for the river Narmada since at least Pauranic times. It occurs in Kalidas's "Raghuvamsha" (R.6.43) and "Meghaduta" as well (Me.19)Here is the 19th stanza of Meghaduta:
All Indo-European languages (Sanskrit and all Indic languages like Hindi, Marathi etc; Iranian languages like Persian, Baluchi, Pashto etc; and European languages like Latin, Greek, French, German, English etc) - are derived from the "Proto-Indo-European language" (PIE).
The word-roots "rei" and "reu" in the Proto-Indo-European language are at the origins of the modern words "river" in English and "rayate" (रयते) - to move, "rinati" (रिणति) - to move, to flow, "riti" (रीति) - moving, flowing in Sanskrit.
The direct derivation of "rewa" from these is not mentioned anywhere - but the sense in which it is applied to Narmada is that of a fast, leaping river. Hence, in my opinion, "rewa" must also have been derived from the same roots.