Why....does the brother want the necklace?

It's something you give to family and loved ones to pray for a safe journey and hope they get back alive. He simply wants to reassure Elise that he'll come back alive from the war.

To “toughen his heart” there’s no guarantee that he’ll survive, it’s a war, but holding on to something that someone precious gave to him makes it bearable as to say. It’s a promise to him and her that he’ll try to come back