Symbolism, Imagery, Allegory
We've all got personal possessions or clothing that define us for our friends and family. Maybe it's your letter jacket from school, the notebook you're always writing poems and story ideas in, or your favorite, worn out baseball cap. Evidently, even with the passage of over two hundred years, some things never change. No matter how well they know him, everyone recognizes Curzon as "the boy in the red hat" (5.33).
As a slave, it would be really easy for Curzon to get depressed or irritated with his circumstances. Even if Bellingham seems to treat him well, knowing that another person owns you and that you have virtually no rights has to be pretty demoralizing.
Still, we never once see Curzon complain. In fact, he seems content and high-spirited as he helps his master by delivering details about the British cause—even when he agrees to join the rebel army in Bellingham's place. Insofar as the red hat represents Curzon, then, it also stands for his hope, spirit, and individualism.
What's really interesting about this hat is that it changes in appearance according to Curzon's experiences. When he declares himself an American soldier, Isabel notices that it's "flecked with mud" (26.27), as though foreshadowing his coming trials as the sole slave in his unit. When she sees the British march him into the prison, it's "nearer brown than red, with a rip through the brim" (33.27)—intact, but worse for the wear.
And finally, when Isabel visits Curzon in prison later in the book and finds that Dibdin has been depriving him of blankets and food, he doesn't even have the hat. Its texture, condition, and color all gradually fade, just as Curzon's own hope for freedom and even survival begin to lose their luster. Isabel's his only hope now.