Your leftover chicken bones can be combined with chopped veggies (such celery, onions, carrots, and bay leaves) and placed in a pot to produce a delicious stock.
The leftover fat and skins, which have already been cooked and seasoned, will add some delicious poultry flavor to your stock, and the bones will add a deeper taste.
You can use the stock to make soup or enjoy it by itself. You'll still have to throw the bones away when you're done, but you'll get more use out of them this way.