Borrow Chieftains albums from friends or a library and you'll find plenty of old airs which would fit the bill, ones which are largely divorced from familiar song settings. "Women of Ireland" was one of the first tunes which popped into my mind, but sadly for your purpose, it immediately reminds one of the movie "Barry Lyndon", and the Chieftains recording of it was played to death, so that it might elicit "not again" groans. I learned "South Wind" from a Chieftains album, too (though I usually play it on concertina rather than fiddle)--sadly, it's another tune so familiar that it might get NAGs.

Another rich mining ground is YouTube. Search for versions of tunes recommended above, and once you find a fiddler you like, check out other tunes posted by the same user or of the same artist. I've discovered lots of hidden gems that way. I recommend that you search YouTube using Google--it stays more focused on what you request than does YouTube's twisted search algorithm--, for instance, with:

site:www.youtube.com lament fiddle
site:www.youtube.com "slow air" [don't neglect the grouping quotes on set phrases]

To keep the searches even more focused, you can try combining with keywords like "selchie", "silkie", "mermaid", "sea", "Aly Bain", "Natalie McMaster" "Kevin Burke"... You'll be amazed what you turn up; my first find was this kid (makes me envious):
Neal Pointon: "Lament for O'Donnell": http://www.youtube.com/watch?v=CS6gtfNSJ1E