The actualy name of the river was 'The Water'? I thought that was just the Hobbits calling it by what it literally is, not the actual name of the River.
I suppose it would have had a different name during the reign of Arnor though? Hobbits don't come around until much later in the Third age, thus I would then assume that once the settled in the Shire they were start naming things anew.
listen, I told you all I could it is a really small detail that Tolkien actually even mentionted such a small river I mean...the Water isn't incredibly big and of no great importance during the history of Arnor also, remember that that area was not inhabited by people as it was empty as the Hobbits came anyway, I personally have found no other name for it
'The Shire-water, full name of stream running down from the North (through Hobbiton and Bywater) and then along the line of the Road to the Brandywine which it joined just above the Bridge' JRRT
'Shire-water' goes well enough with the name of the 'Brandywine', Branda-nîn'Border-water'.
Adding a bit for interest maybe: the river names Axe, Exe, Esk and Usk are all derived from the British word isca meaning 'water'. Source Reaney, P. H. The Origins of English PlaceNames.
