Some characters in plays and novels simply aren't rich, fully developed characters. And that's OK – they might serve another purpose. Rahim Khan, we think, serves as the novel's moral center. If Hassan and Ali are off in the land of bright, shining moral purity, and Assef is in depths of devilish cruelty, and Amir and Baba are somewhere in between, Rahim Khan is a voice reason standing outside this hubbub of moral questing. He's kind of like Horatio in Hamlet – you couldn't base a whole play on him, but you're glad he's there because he makes you feel sane.
Come to think of it, Rahim Khan is literally the moral center or voice of reason in The Kite Runner. He calls Amir in California and flat-out says: "There is a way to be good again" (1.3). He also functions as a second parent for Amir. When Baba ignores Amir, Rahim Khan is right there with an encouraging word. ("As always, it was Rahim Khan who rescued me" (3.43).) Perhaps, too, Rahim Khan helps expand Amir's understanding of ethnicity. Rahim Khan tells Amir a story about how he almost married a Hazara woman. His family reacted very strongly (death threats!) and Rahim Khan never married the woman, but the story lets Amir know that someone else he admires thinks of the Hazara ethnicity as equals. (Amir's mother would be the other someone.) All this confirms that Rahim Khan is a stand-up guy.