What does Grendel's Mother do to Beowulf to show that she is bent on revenge in the poem?
Grendel’s mother avenges her son by attacking Heorot at night, killing Hrothgar’s friend Aeschere, and taking with her the trophy of her son’s dismembered arm. Unlike Grendel, the mother’s motivation seems purely retaliatory; she is less interested in plunder than she is in making the Danes pay. This is why she leaves such a clear trail behind her when she returns to her lair in the lake, and why she marks the spot for Hrothgar and Beowulf with Aeschere’s severed head. She is luring Beowulf into the lake, where she hopes to have a final revenge. Beowulf is more than up to the challenge, however. He jumps in the lake, and is right away captured by Grendel’s mother, who takes him to her underwater home. There is a savage fight, during which the sword Unferth gave Beowulf fails; luckily there is a huge sword, meant for giants, that is part of the mother’s armory, which which Beowulf finally kills her.
The death of Grendel’s mother is one of the most violent parts of the poem. Unlike Beowulf, who seeks glory, Grendel’s mother only seeks vengeance for her child. After Beowulf kills her, he uses the giant sword to decapitate Grendel's corpse, which is in his mother's lair, and brings the head back to Heorot as a trophy. One can see how a reader might sympathize with the grieving monster in this situation.
The first act of revenge is her journey to Herot to retrieve her son's arm which was placed in the rafters as a sort of trophy after Beowulf ripped it from Grendel's socket. While she was there, she killed and ate Hrothgar's best friend.
Beowulf, in answer to the death of Hrothgar's closest friend, traveled to the bottom of the bubbling lake to fight the creature. While Grendel's mother is bent on revenge to avenge her son's death, she honorably protects Beowulf from other sea creatures and also takes Beowulf to her underground lair filled with air which puts the two on somewhat equal fighting ground. However, the honor ends there. She gnashes and tears at his armor, breaks his sword, stretches him out on the floor of the cave, and fights ruthlessly in order to kill her son's murderer. If it weren't for the magic sword forged by giants which hung on her cave wall, Beowulf may not have won the battle.
It is with this sword that Beowulf kills Grendel's mother and decapitates Grendel's body in order to haul the head back to the mead hall for one last hurrah.