Carmen's character is all about discovery and exploration, and the tragic way things can end if circumstances are stacked against you. Don't start getting paranoid and thinking the glass is half empty, though: that's a theme that a lot of operas have, so it makes sense that her name means "song." Oh, and Carmen is also the name of a doomed opera heroine, so that might give us a clue that things ain't gonna end well.
Carmen discovers lots of things over the course of the book: friendship with Roxane (6.3), the beauty of music (5.205), and love for Gen. Even the experience of visiting a fancy house is new to her and seems full of wonder (5.205). And Carmen's pretty cool, which makes it easy to be excited for her as she discovers all these new things. Even if she carries a gun. Once she puts it down, she seems intelligent, kind, and likable.
Carmen not only discovers new things herself, she's also the reason Gen discovers a whole new world. And not just on a magic carpet ride (they prefer the china closet, after all): when Gen falls in love with her, it's a completely new experience that changes his life deeply (see his character analysis for more on that).
But as in opera, where passionate love and new discovery all too often end in tragedy, Carmen's circumstances mean she's doomed from the beginning of the book. She can't transition into the world that Gen inhabits. Why? Because she's been part of a widely publicized terrorist takeover. Hard to talk your way out of that.
It's easy to see it as unfair for the outside world to judge her for this. She's just a teenager, and she's possibly the least violent of a particularly moderate and idealistic group of terrorists: they don't kill anyone in the entire book and Carmen doesn't even like to point her gun at people, even if she carries it around at first (10.17).
Plus, it's pretty clear that once she realizes there's an alternative way of life, she's open to leaving the terrorists' lifestyle behind altogether. She seems relatively innocent, and it's likely that a combination of poverty and persuasion by the Generals led her into a terrorist lifestyle she wouldn't have chosen if she'd had more options.
But none of that matters to the government forces who gun her down. Like so many passionate and relatively innocent heroines of opera, she comes to a tragic end. Is the passion and exploration she found in her time in the house worth the tragic price she pays? It's another one of those questions Bel Canto leaves us thinking about.