What is the name of Odysseus' dead mother and how did she die?
The name of Odysseus' dead mother is Anticleia. She dies of a broken heart.
Accordingly in Book 11, Odysseus makes the the journey to the Underworld, the Kingdom of The Dead. One of the first ghosts he sees is Elpenor, his former friend and companion. Elpenor begs Odysseus not to leave him unwept for and unburied. He asks Odysseus to burn his body in full armor and to make a burial mound for him so that men will always remember how his luck ran out.
Eventually, Odysseus catches a glimpse of his dead mother, Anticleia. He is shocked, as she was alive when he last saw her. Seeing her here in the Underworld causes him great grief. He decides to speak to Tiresias, the prophet, before he approaches his mother. Tiresias himself approaches and drinks the blood of Odysseus' sacrifice. He tells Odysseus that his ship will be destroyed and his men will die on his journey home. However, he also lets on that Odysseus will kill all Penelope's suitors when he gets home and that when he is old, he will die a gentle and painless death.
At last, Odysseus is ready to speak to his mother's ghost. She approaches and drinks the blood of the sacrifice. Upon drinking, she recognizes her son and is greatly grieved. She questions his presence in the House Of Death, telling her son that it is a terrible thing for living men to catch a glimpse of the world of the dead.
Odysseus tells her that he was forced to seek the counsel of the prophet, Tiresias. He entreats his mother for news of his father, his son, and his wife. At length, he begs his mother to tell him how she died, whether by illness or by Artemis' 'painless shafts.' She asserts that it was neither. Instead, she died of a broken heart, of great longing for her son.
No sharp-eyed Huntress showering arrows through the halls/ approached and brought me down with painless shafts,/ nor did some hateful illness strike me, that so often/ devastates the body, drains our limbs of power./ No, it was my longing for my shining Odysseus—/ you and your quickness, you and your gentle ways—/that tore away my life that had been sweet.