Bilbo gives the Arkenstone of Thrain, 'the Heart of the Mountain', to Bard to aid him in his bargaining with Thorin. Bilbo recognizes the importance of the jewel, telling Bard that "it is also the heart of Thorin. He values it above a river of gold
Answer: Bilbo gives the arkenstone to bard because he believes it will assist him when bargaining with Thorin.