I agree that a lot of the problem with many of the songs is that they're overplayed or done badly. Unfortunately, that tends to ruin the sweetness of getting the occasional taste of a song done well. I was at the Winterguard championships in Wildwood, NJ this past spring and saw a guard perform to a beautiful version of "Danny Boy".. since it WAS well done and I hadn't heard it recently, it DID bring tears to my eyes.
My submissions:
The Ball of Ballynore (a.k.a. The Gathering of the Clans): Most people probably don't hear this often, but at a Renaissance Festival it's likely to get you set upon by every kilt-boy, actor, and volunteer in sight because we've heard it so damned much! However, a new verse can still occasionally be welcomed with a laugh.
Come Forth: I don't know how widely known this one is, but it's a "Here Comes the Bride" type of thing. "Come forth, come forth from Lebanon, my bride; Come forth, come forth from Lebanon, my spouse; Come draw near, come draw near, my fair one." etc.. I made my mother promise that if/when I get married, she'll make the choir NOT sing it for me, otherwise I'm going to refuse to walk down the aisle. :)
99 Bottles of Beer on the Wall
Amazing Grace on the bagpipes: I never heard this until maybe last year, and suddenly it's all that the pipers know how to play! Now, since I hate hearing a good song mangled, this was occasion for fist-clenching, frantic whimpering, and serious thoughts of setting my clan-cousins on the pipers. However, has anybody heard Ani DiFranco's version of AG? It's beautiful because it's done in a different style.