What quality in Ivan does Gopchik bring out?

He brings out Ivan's ptotective, fatherly side. Ivan likes the boy and help him.

that would be protective...... sorry!

Ivan Denisovich was fond of Gopchik, "the rascal (his own son had died when he was little". Ivan sees Gopchick as resourceful and quick thinking. The boy brings out Ivan's feelings for his own family. Ivan does care for him although Gopchick is quite resourceful at taking care of himself.