In Gaelic, 'Ard-na-Saighead', with the last word pronounced roughly as 'sate', means 'Height of Arrows'. 'Arthur's Seat' is probably just a corruption of that and there is some evidence of an Iron Age fort on the hilltop, so the description fits.
It is also possible that it might relate to King Arthur, who - some legends suggest - came from Scotland. Such a site would have been considered suitable as the 'throne' of a hero of such stature.