A bone is an unsellable (though droppable) item with unique purposes in both Golden Sun and Golden Sun: The Lost Age. In the first game you can get an unlimited quantity from a girl in the inn at Vault, but you would only have use for one. Give it to the dog at the elevated bottom portion of the town and it will move up and give you a hint via Mind Read that the entrance to the optional dungeon Vault Cave is right here and can be entered if revealed. A player who knows where the entrance is wouldn't even need to do this step because you can get to the entrance without the need for the dog to move anywhere.
In the second game, a dog in Lemuria hints that there are a lot of items buried underground that can be acquired with Scoop. The nearest spot containing a Scoopable item contains a Bone, which in this game is completely useless. Its existence is meant to disorient players into believing that the bone is the only thing the dog was referring to as a hidden item; players who search harder with the help of the Reveal Psynergy will find that there are, in fact, other items buried under the soil, including a Lucky Medal and the Corsair's Edge as a rusty item. There is only one Bone in The Lost Age, so if dropped there is no way to acquire another. Of course, since Bones have no practical purpose in The Lost Age, there is no great loss in disposing of it.