After Beowulf defeats Grendel, ripping off the monster’s arm in the process, the Danes celebrate the end of Grendel’s depredations by hanging his severed arm in Heorot. However, their celebration is short lived. The next night, Grendel’s mother enters Heorot and kills Aeschere, one of King Hrothgar’s most trusted advisors. She then takes Aeschere’s body and Grendel’s arm from the king’s hall. Thus, she deprives the king of his closest friend and the Danes of their grisly trophy.
The main motivation attributed to Grendel’s mother in the poem is that of vengeance for the death of her son, although the poet also provides us with some information that her sorrow at the loss of Grendel is also a motivation:
1255(b) That became manifest,
widely known by men, that an avenger still
lived after the misfortunes, for a long time
after the war-trouble, Grendel's mother,
lady troll-wife, remembered misery…
1276(b) And his mother even now,
greedy and gloomy-hearted wished to go forth,
a sorrowful journey, to avenge her son's death…
1302(b) [S]he took from its gore
a well-known arm; sorrow was renewed,
it returned to their dwellings;
Given that Grendel’s mother comes to Heorot both to avenge her son and because she is grieving his loss, it seems likely that these two emotions also combine to motivate her to take his arm back to their dwelling. She wants to deprive the Danes of their trophy, but it also seems that she wants to reunite Grendel’s arm with the rest of his body. This indicates that Grendel’s mother, though a monster, is able to feel love for her son and grief at his loss, to the degree that she wants his entire body back.