In Iliad 7, whose idea is it to call a truce and have Hector challenge an Achaean to a duel and why is the duel suggested?
Iliad 7 opens with Paris/Alexander returning to the battle alongside Hector. In Iliad 3, the duel between Menelaus and Paris had been unable to bring an end to the war, so in Iliad 7 another attempt is made to bring the fighting to an end. With this duel, however, it appears that only the fighting for that day will come to an end. Thus, Apollo says,
it would be best by far to end this day of strife and conflict. Let them fight another day with Ilium at issue... (A.S. Kline translation)
In Iliad 7, Athena, who favors the Greeks, and Apollo, who favors the Trojans, decide to inspire Hector to challenge on the Greeks to a duel similar to the one fought by Paris and Menelaus in Iliad 3.
This divine inspiration is transmitted to the mind of the Trojan Helenus, who "divined in his mind this plan, in which the immortals had concurred" (A.S. Kline translation). Helenus relays this plan to Hector, who is happy to issue the challenge since Helenus also tells him that the gods have declared that "it is not your fate to die, your time has not yet come."
Thus, Hector realizes that he could bring some end to the fighting and he could also remain alive.
The gods and goddesses of Olympus become involved in the Trojan War and frequently attempt to influence the outcome of events. In this instance, it is Apollo (who favors the Trojans), the god of sunlight and prophecy and the wise goddess, Athena, supporter of the Greeks, who confer with each other and decide to bring the war to a halt by calling a truce. The plan is to have Hector, the greatest Trojan warrior do battle with a chosen Greek warrior, and that both sides will abide by the outcome. If the Greek warrior dies, the Greeks will abandon the battlefield and return home; if Hector falls, the Trojans will withdraw behind the walls of Troy and life will go on. Both sides agree to the truce and the Greek warrior, Aias (Ajax) is selected to fight Hector. The truce is arranged because both the mortals and the immortals are eager to end this terrible war.