Long ago I was told Binnorie is a corruption of By Norham and this certainly fits the feel of the ballad, especially as given by Bruce & Stokoe in The Northumbrian Minstrelsy (1882) which has a melody to die for. Whilst they do point out it ubiquitousness throughout Europe and Scandinavia in my mind when I sing it The bonny mill-dams of Binnorie are ranged along the Tweed beneath the soaring ramparts of Norham Castle.