Ormond Castle was named after the hill it stood on, Ormond Hill. It is now impossible to trace how the name came about, but the Scottish antiquarian John Pinkerton says it was apparently an ancient moot-hill. Incidentally these were known in Scottish Gaelic as
tom a' mhòid, which may provide a clue as to Ormond's etymology.
[I]t appears that Ormond was a hill, (probably an ancient moot-hill or court of justice,) in Ardmanach on the east of Ross-shire.
Pinkerton, John. The History Of Scotland From The Accession Of The House Of Stuart To That Of Mary: With Appendixes Of Original Papers. In Two Volumes. Dilly, 1797.
The castle itself was actually known by several other names, such as "Castle of Avoch" after the nearby village, or "Douglas Castle" after its owners. Hence it was the name of the hill, not the castle, which provided the source of the Scottish peerage titles of Ormond.
In 1481, as we have seen, King James III. granted the lands of Avauch, with the moot-hill of Ormond, to the Marquis of Ormond, who about 1503 resigned the lands, but retained the moot-hill in order to preserve his title.
Stell, Geoffrey. "Architecture and Society in Easter Ross before 1707." JR Baldwin, Firthlands of Ross and Sutherland (1986): 98-132.
Apparently it later became known as "Ladyhill" because of a chapel to the Virgin Mary.