The courage shown by Hector and Andromache lies in their refusal to abdicate their duty. Both characters are shown to be victims of the duty that is placed upon them by the gods or by society. Yet, they are not victimized by this conception. Both of them rise and display the very struggle in what it means to be human. It becomes clear that Homer identifies and cares for both of them because of the courageous stance both take. This is primarily because of their refusal to surrender duty and sacrifice honor, even if it comes at grave and painful costs.
Andromache is the prototypical wife of a soldier. In comparison to Helen, Andromache's courage is almost an ideal. She recognizes the her desire to prevent her husband from fighting is a primary motivation within her. Yet, she has the courage to recognize how important Hector is to the cause of Troy. She has the courage to not push herself into a position that makes his position even more difficult. She shows courage in the care she gives to her son and to her husband. She recognizes that he will probably die, and yet shows courage in not dwelling on this.
Hector's courage is evident in nearly every mention of the epic. He does not waver in facing what it is he must do. When his son fails to recognize him in his battle armor, it is a moment where the lack of courage could compel him to run from what he has to do. Yet, he shows courage in honoring his son with a prayer to the Gods. When Achilles is ready to fight him, Hector shows courage in facing what must be done. His willingness to honor Achilles, even in suggesting that he will give him a proper burial should the Trojan kill him, shows Hector's courage to face death and not retreat in the face of challenges that would cause others to do so.