Randolph is a rowdy (and even annoying) little kid. He is symbolic of loud Americans in general.
Randolph lets Daisy know that Winterbourne isn't uptight, like other Europeans. Therefore she is likely to like him, since he is more relaxed.
Later, he influences the action because Mrs. Miller cannot go out with Daisy and Winterbourne because she is struggling to get Randolph into bed. For this reason, Daisy and Winterbourne go out unchaperoned, definitely something scandalous by European standards.
