If a composer could say what he had to say in words he would not bother trying to say it in music. -Gustav Mahler
Music is a uniquely human form of communication that has its own rules of grammar and punctuation. It can be beautiful and haunting, or it can be uplifting and spiritual. Most people seem to agree that it's an important and affecting artform. A good example I can think of for Zelda fans is the opening of The Windwaker. How can you help but cheer when the original Zelda theme starts playing?
Sheik's primary role in Ocarina of Time is teaching Link songs. Sheik plays the harp, though it's clear that she knows how to play the ocarina as well. The harp is a more familiar instrument to western audiences (when I heard that the new Zelda installment would be called Ocarina of Time I thought it was the funniest thing ever Ocarina! Hee!) Anyway, the harp is a very ancient instrument, and has a heavy role in folklore. King David had a harp, and their music was credited with supernatural powers that could "destroy the feynde's myght."
It is also an instrument of medieval romance. According to the 12th century Laws of Wales stated that the three things a gentleman should never be without were "his harp, his cloak, and his chessboard." It was the instrument of wandering troubadours, who used their melodies to accompany their songs of unrequited longing and courtly love.
While Sheik doesn't exactly function as a troubadour (she's a troubadour NINJA!) there is an element of the wandring minstrel to her character. The melodies she plays have powerful effects, and it stands to reason that she knows more than she taught Link anout the ways of music. In the Zelda series music has an important role, which I think serves as a way to elevate the story to legendary status. The songs in Ocarina stay with you (sometimes for far to long) and offer something universal among Zelda fans. No matter what language you played the game in, you heard the same Zelda's Lullaby. By aligning Sheik's power with that of music it increase both her magic and her mystery.