Click for Part 1 (Snow White), Part 2 (Cinderella), Part 3 (Aurora), and Part 4 (Ariel).
By default, I started answering that Belle was my favorite Disney princess because she, like me, has brown hair and brown eyes and she also reads. And I suppose, when I was younger, I did relate to her sense of longing, a longing for an indefinable something.
Like Ariel, Belle came from what we would consider a more modern age, as compared with the 30's or 50's. Two years after The Little Mermaid, Beauty and the Beast came out in 1991 (which also happens to be the year I was born, which further links me with this film). So Belle shares some of the feistiness of Ariel. For instance, she doesn't feel compelled to accept Gaston, she defends her father against Gaston and LeFou, and she talks back to the Beast when he shouts at her. Further, though, I find that Belle's feistiness, if that's what we're calling it, is grounded better than Ariel's teenage rebellion.
For one thing, Belle is a little older than Ariel. While Ariel is a teenager, Belle seems to be more of an adult (technically, it seems that Belle is only 17, but that still makes her older than majority of the princesses). She picks her battles better than Ariel does, as well: she defends herself and she defends the people she cares about (let's take a moment to appreciate Belle's good relationship with her father). And for the first time, instead of simply stressing good virtues, this story introduces the idea of intelligence. Belle reads constantly and her father is an inventor, so she is used to thinking about various ideas (note that when her father asks her to hand him a tool, she knows it by name--therefore we can assume that she is familiar with the work he does). This is in fact the most significant contribution Belle gave to the princess group--after all, if you want the princesses to make good/admirable decisions, then they have to be characters capable of making those decisions.
Belle's weakness goes hand in hand with her strength. Because she is a thinking person, she is also curious--and sometimes that curiosity takes her too far. With her head so stuck in novels (similar to Catherine in Northanger Abbey), she wants to see what's going on in the castle and so ventures into the forbidden West Wing. I can never forgive Belle for this. She has free range of the entire castle except for this one section (which can, and in fact is, simply be the Beast's quarters), and yet she just has to go snooping over there. Still, it worked out in the end: her argument with the Beast led to their making up and trying to be nicer to each other. And what would a character be without some faults?
I complained, in my analysis of Aurora, about how one of the gifts from the fairies was beauty. So perhaps I should also be complaining that so much is made of Belle's looks? But this brings us to an interesting topic. Consider who talks about Belle's looks and in what context. Primarily it's the villagers: they bemoan the fact that Belle is so good-looking and yet so odd (remember, her name does mean beauty and many English translations of the story also choose to translate her name from the French Belle and call her Beauty--so the film had to address her looks, one way or another). Gaston also talks about how Belle is the most beautiful girl in town and that's why he wants to marry her. So the obsession with looks is paired with triviality, and there is also a comparison made between how someone looks and how they act. Even LeFou basically tells Gaston that, yes, Belle is beautiful, but isn't her personality ill-suited for you? Belle sees it--she tells her father that Gaston is "handsome alright" and yet "he's not for me." This, of course, in turn leads us to the theme of the movie--the whole "beauty is found within" concept.
Now, then, I have one more big issue to try and cover in a short space: the fact that Belle falls in love with her temper tantrum captor. Here is also the part where I start rolling my eyes because I think that people make far more of a fuss about this than they need to. Beauty and the Beast doesn't encourage abusive relationships; it encourages intelligence. Here is what's really going on. A sorceress decided to teach a spoiled prince a lesson by putting a spell on him that would only break if he could overcome the concept of looks. He looks like a beast but he has to act kinder than he did when he was a (probably good-looking) boy/young man. Belle eventually has enough generosity to give him a chance (remember, she knows from the start that there is a spell--she doesn't know exactly what it is but the fact that the furniture talk shows that there is something going on--and the fact that Belle is used to reading fantasy stories prepares her for possibilities). And he does take this chance: he lets her go. You heard that, right? This means that when Belle tells the Beast that she loves him, she is not his captive--she has instead chosen to return to the castle to protect him from Gaston and the angry mob. He, too, chose to turn his back on the past by literally turning his back on Gaston instead of fighting him back.
So the lesson that the Beast had to learn (to look at the person inside rather than the outer shell) is a lesson that Belle shared in--only it took her a few days or weeks (I'm always confused about how much time passes in the movie) instead of several years like it did for him. And for those who complain about Snow White or Aurora's lack of agency, Belle has agency. The Beast fell in love with her first, making the declaration to his servants after he let her leave the castle. But it is Belle's declaration that is the last piece, the piece that breaks the spell and frees him.
I'm therefore naming Belle as one of the most well-rounded of the Disney princesses.