
Because this is totally better than actually working on my paper due in four hours:
There is no reason for Achilles to refuse Agamemnon's gifts in Book Nine. By giving Achilles gifts, Agamemnon admits that he needs him, and that he was wrong to take Briseis. The gifts, by the way, are worth much more than the slight merits - (from memory) seven tripods [and one tripod, as we learn from the contest, is better than a woman], seven Lesbian women as well as Briseis, one of his daughters to wife, and seven cities. (I think it's a feature of Agamemnon's character that he goes too far in everything: his wrath against Achilles was excessive, but so are the gifts he is willing to give in order to appease him.) Further, Agamemnon subtracts from his own kleos by giving up his own prizes, which the Lesbian women are. He's admitted that Achilles is greater than him and he is willing to lose kleos, which is what Achilles wanted - for Agamemnon to honor him. There's no point in asking Thetis to ask Zeus to destroy the Argives if he's not going to go into battle (unless he actually wants all of them to be killed, which I doubt he does).
Since Achilles has what he had prayed for, the only reason for him to refuse is plot. If he went back to the war at this point, Patroclus wouldn't have to go for him, and would'nt be killed. But the only reason for Patroclus to die is for Achilles to go back into the war, which, if he had gone already, wouldn't need to be forced.
Sure, Achilles states reasons for not going back into battle: because he thinks that Agamemnon is still taking too great a share, because he himself has no quarrel with the Trojans - but then why doesn't he go home? One could argue that it's because he does want to fight, and that his reasons are really excuses, but then why doesn't he leap at Agamemnon's offer?
Depending on how you feel on the authorship of the Iliad, it's a case of et annuit Homer or of a place you could point to as evidence for multiple authors. Say the scene of the embassy is in one version, and the compiler[s] liked it very much and wanted to keep it, but also liked the Patroclus situation we know and love. Perhaps in a different version Achilles took the offer and got killed, as was his fate anyway.
Whether or not it points to multiple authorship, I do think it's interesting that Achilles doesn't take the offer. Maybe I'll write my extra credit paper one that.