I thought that Lanayru Mining Facility's music was one of the best songs in the game and I listen to it often. When I first heard the song the tinkling sound irritated me too, but as I listened on I realized how much there was to the song and why that strange sound was good. It complemented the song as a whole quite well and it helped to give the dungeon its distinct atmosphere.
Lanayru Mining Facility was an old ruin, so the song had that odd hollow tinkling and often deep wind-like sounds, but the place was also one of advanced technology, and the high-pitced sounds and various other aspects of the song work with this theme quite well. Furthermore, the dungeon was also highly dangerous. This was the climax to the game's first half, as well as dungeon of Lanayru Desert. Lanayru Desert was a sad area that, depending on how you want to interpret it, may have actually depicted portions of the Demon War talked about in the opening (the events that seem to be going on in the past when you activate Timeshift Stones look like they're from that war). Lanayru Desert had an adventurous atmosphere, but also a sad one. Lanayru Mining Facility was the dungeon of that area, the dangerous portion, so they offset that with an unsettling and alarming tune. In addition to the themes I already addressed, the Lanayru Mining Facility theme was supposed to be uncomfortable. It was a vaguely irritating noise for me, but not in the sense that it made the song difficult to listen to, but in the sense that it made the song unsettling. This fit the area -- with its danger, creepiness, and sadness -- very well.