Little Randolph Miller has his sister's spirit in a rougher, even more immature, masculinized form. He's loud and brash and doesn't want to do anything that anyone else wants to do.
Daisy claims Randolph is quite smart for his age (nine), but we're given no clear evidence of that. Instead, his characterization generates the impression that he's convinced his mother and his sister to let him have his own way (lots of candy, no bedtime, little supervision) through intimidation and empty bravado.
His rough, physical energy and the fact that he's poking things with a large stick when we first meet him have caused some readers to assert that he's a walking representation of sexual desire. You can agree with that or not, but either way, he's highly symbolic. He definitely symbolizes American patriotism, because he is the constant champion of America's superiority. He practically spends the whole novel chanting "U-S-A! U-S-A!" In his comparisons of Europe and America, Europe always falls short.
To sum it up: "'I don't know,' said Randolph. 'I don't want to go to Italy. I want to go to America'" (1.39).
You might get annoyed by "America this, America that" when the little bugger is on a year-long candy binge and luxurious trip through Europe, but remember that he's been practically abandoned by his father and neglected by his foolish mother and teenage sister. Then maybe you'll be able to muster the appropriate sympathy.
Randolph likes America because his dad's there. But his hyperbolic patriotism also serves as a reminder of the kind of ignorant pride about a cultureless country that James thinks Americans are guilty of. So maybe it is kind of annoying after all.