The music box belonged to Danielle Rousseau and was first seen in the episode "Solitary", whilst Sayid was held captive in Rousseau's camp. The box was initially broken, but Sayid managed to fix it, much to Rousseau's delight.
The music box can be seen as a metaphor for Rousseau's psyche as she had been "solitary" for so many years. Much like the box, she had been broken by Ben. By speaking with someone for the first time in years, she finally had contact and was able to heal somewhat. This is supported by the fact that when Sayid asks to be let go (ceasing contact), the music box stops playing immediately as Rousseau says "Go?" in a heartbroken tone.
The fixing of the box seems to create a bond between Rousseau and Sayid, which will later lead to Rousseau being more willing to interact with the survivors of Flight 815.
When the box is fixed, Rousseau smiles. This is the first, and one of the only times in the show that she has made this expression.
The song that plays in the box is "Intermezzo" from Carmen (Georges Bizet, 1875). A small dancing pair is seen twirling in the center.