It seems the name of this promontory fort might come from the mythical Aine. The stone mentioned below couldn't (can't?) be far away - surely someone knows more about it.
.. a great stone called "the chair of Aine, or the chair of the lunatics," was located, possibly still is, near Dunany; and the people generally believed that lunatics, actuated by some insuperable impulse, if at liberty, usually made their way to this stone, and seated themselves thrice upon it; and it was generally believed that after having performed that ceremony they became incurable. It was also considered a very dangerous act for persons of sane minds to sit upon this stone, lest they too might become subject to the power of Aine, that is, become affected with lunacy.
The human race were not the only beings supposed to have been affected by the mischievous Aine, since rabid dogs even were said to have come from many parts of the country and flocked around this stone, to the great danger of the neighbours and their cattle: when they remained around the lunatics' chair for some time, they then retired into the sea, as if compelled by some potent invisible power, and the people supposed that they were forced to visit the submarine dominions of Aine, since they were entirely under her subjection.
Aine is said to be connected with the moon, which seems apt if she has a chair for 'lunatics' and has connections with the sea.
Quote from an article on Folklore by Mr. Nicholas O'Kearney, in 'Transactions of the Kilkenny Archaeological Society' vII (1852-53), on p35. (you can see it on Google Books).