A jury usually withheld names in order to remove any effects of names, castes etc into the process. Inside the jury room, the people are simply humans trying to impart justice. When two people swap names, it signifies a bond, especially if it is done after passing through an experience together. The men, after having gone through a emotional, social and philosophical awakening inside the room, share a common realization of the darkness inside every one of them. They have now formed a bond because they believe each of them has contributed to the right by voting not guilty. When the two men exchange names, it is their belief and respect in each other, as well as the implication of a beginning of a bond or friendship between the two. When (in the old times), strangers came to town, the names would be exchanged only when they had gathered respect or became enemies. Thus, the name exchange signifies the human social element in the men, that outside the jury room, they are just ordinary people in their lives.