There are various interpretations of why the old Raoul brings the monkey music box to her grave and some even think that all the old Raoul parts of the movie are completely useless. However, I'm one of those who believe that while the film would've worked perfectly well even without Raoul travelling anywhere with the music box, all the old Raoul clips are perhaps the most essential aspect of the entire film's basic point. I think they are the ultimate aspect that drastically differs the film from the stage play.
A monkey symbolizes many things that are strongly related to the story. Essential to note is that the movie's end has a moment where Christine is sailing away with Raoul but turns to look back at the Phantom, while she sings the words "Share each day with me, each night, each morning..." That moment is also in the original screenplay, which also states that she seems to be singing that to the Phantom.
In India, the monkey is a symbol of soul. Monkeys may represent lust. A monkey may also represent positive attributes, such as inspiration and a sense of freedom. Monkeys as symbols appear to have a double or twin meaning. Whether positive or negative, the monkey is revealing something about what is going on in your inner world.
This is why it may be validly interpret that Christine, though had been a happy and beloved wife and mother in her marriage with Raoul, she had always been torn in two. As in never felt truly free because by leaving the Phantom behind, she parted from an essential part of her. The Phantom had once inspired her voice and she obviously had felt tons of lust towards him and according to the film's director, Joel Schumacher, she felt a very deep and very soulful union with the Phantom which can be interpret from the end when she seems to be singing to the Phantom those certain lyrics.
Perhaps, by bringing to her graveside the monkey music box which had belonged to the person with whom Christine had emotionally/spiritually shared every moment of her life, Raoul wanted to make a gesture of understanding or apology to her.
After all he went through a lot of trouble to get the music box and the film makes it seem like the story we see is him remembering the events on his way to Christine's grave.
From the very first view in the movie theaters almost six years ago, I personally interpret that Christine deeply loved the Phantom and wiseversa, perhaps a little even in a romantic way because he had showered her with red roses and she looked so deeply moved by his very romantically done All I Ask Of You Reprise in the end of PonR... But for whatever reasons, they didn't get to share a lifetime together... yet they shared a spiritual bond every moment of their life like she sang to him that she wishes them to... and then that the final scene at her graveside suggested that she'd been torn in two and at least finally she and the Phantom could be together - in eternity. The Phantom's rose and engagement ring there only supported my interpretation.
I believe all this makes a lot of sense, because Andrew Lloyd Webber gave the direector, Joel Scumacher, very free hands to make the film in his own view and that he, ALW, feels that Joel kept the essence of the stage play but added a deeper emotional center. And, Joel himself has confirmed there was love from Christine to the Phantom, and on the official web site of the film, Joel has said that what drew him into directing The Phantom of the Opera, was most essentially, the tragic romance between the Phantom and Christine.