In an article of the Canadian Journal for Traditional Music, 1975, the author says it means "forever" in the River Shannon song: click here

Also, found this:

Far more precious (but perhaps no easier to catch) is a different sort of treasure, the kind embodied in the word asthore. An Irish term of endearment, asthore comes from the Irish Gaelic a stor -- literally, "oh treasure."

Also say a lot of use as a surname and place name. Now, maybe those across the pond will wake up soon and tell us which is correct!*bg*

kat