Technically, of course, Orestes is on a mission to kill Aegisthus and Clytemnestra, but he never seems to give much thought to killing Aegisthus. He doesn't have very much respect for the guy and, frankly, no one else seems to either (including most audience members and readers). Killing Aegisthus will be a piece of cake; the real challenge is facing down Clytemnestra. This isn't just because killing one's own mother is a terrifying and horrible thing to do, making the killing of Clytemnestra uniquely emotionally challenging; it's also because Clytemnestra is an immensely intelligent and resourceful woman, who goes toe to toe with her son in their final confrontation, leaving no stone unturned in her quest to escape. It isn't for nothing that, when we first hear that Aegisthus has been killed, Aeschylus makes Clytemnestra call out, "Someone give me an axe to slay a man, at once!" (889).