The precious things mentioned in the story are the leaden heart of the happy prince and the dead bird. They are precious because both the happy prince and the swallow were very kind, generous and selfless. The prince could not bear to see the ugliness, misery and suffering in his city and so gave away all his precious stones and gold to make his people happy. This inspired the swallow to sacrifice his trip to Egypt.
He stayed back and acted as the prince’s messenger carrying the precious stones and gold to the needy spreading happiness around. As the kind prince was blinded, having given away the sapphires of his eyes, the swallow, out of love chose to not leave him. By now it was getting colder and colder with winter approaching rapidly.
Finally, he could no longer bear the cold and died at the feet of the very statue that loved him so much. This broke its leaden heart into two. When God asked one of the angels to bring him two precious things in the garden of paradise, the angel brought the leaden heart of the prince and the dead bird. God proclaimed that in his garden of Paradise the little bird shall sing for evermore and in the city of gold the Happy Prince shall praise God.