Genesis 44:20
And we said unto my lord, We have a father, an old man, and a child of his old age, a little one; and his brother is dead, and he alone is left of his mother, and his father loveth him. And we said unto my lord, We have a father, an old man, and a child of his old age, a little one; and his brother is dead, and he alone is left of his mother, and his father loveth him.
Benjamin was Jacob's youngest son, a little child of his old age. His brother Joseph was believed to be dead. He alone was left to Jacob by Rachel, the mother and beloved wife whom Jacob loved. Jacob himself dearly loved Benjamin.
Jacob's life was bound up with the lad's life. Jacob had been reluctant to send Benjamin to Egypt. Judah explained that to separate Benjamin from his father and lose him would bring unbearable grief upon his aged father, bringing him down to grief and death.
Moved by this, Judah offered himself as a slave in Benjamin's place so that Benjamin could return to his father.