What is there to the music box? It's just "a little square-shaped object […] painted with roses and lilies of the valley. It was the one pretty thing [Mae] owned and she never went anywhere without it" (2.20). It doesn't have any special powers, and it sure won't make you immortal. But that doesn't mean it's forgettable.
The music itself is something of a staple in Treegap. When Winnie first hears the music, her grandma exclaims, "'Did you hear that, Winifred? That's it! That's the elf music I told you about. Why, it's been ages since I heard it last. And this is the first time you've ever heard it, isn't it? Wait till we tell your father!'" (4.24). Hmmm. Why would Grandma think the music came from elves? Probably because it's mysterious—just like the Tucks.
In fact, when Yellow Suit Guy hears the music for the first time, he knows that the story he's heard for so long is true. Bottom line: the music is the one thing the world knows about the Tucks, and it reminds us that they're out there in the world, even if they're always hiding.
Mae and the Music
The box itself—that pretty little object—might also represent Mae's goodness. Think about it. When Winnie is doubting if she should trust the Tucks, Mae hands her the music box.
Winnie turned the key. It clicked faintly. And then, after several more turns, the music began to play again, brisk from its fresh winding, and merry. No one who owned a thing like this could be too disagreeable. Winnie examined the painted roses and lilies of the valley, and smiled in spite of herself. (6.20)
After holding it in her hands and listening to its music, Winnie makes the call: the Tucks are good people.