"Avon" derives from the British language abona, "river", which also survives as a number of other English and Scottish river names, and as modern Welsh afon (pronounced [ˈavɔn]), "river".
The English name for the river derives from its Celtic name Usa, from *udso-, "water," which derives from the Indo-European root *wed-, "wet, water" (and the same root from which we derive water and wet). Thus the Ouse River etymologically is the "Water River" or the "Wet River." Of course, the English who borrowed the name from the Celts did not know the meaning of the wordas is rather frequently the case when foreign topographical terms are borrowed.
At the same time the river names Axe, Exe, Esk and Usk are all derived from the British word isca meaning "water".
It would seem none of the naming was very original to begin with.