Make a little extra cash.
If just the sight of the wedding ring makes you feel ill it may be best to just sell it. Use the money that you get for it to buy something just for you. Depending on how much your ring is worth, you may even want to use the money you receive to put a down payment on a new car or new home for yourself. This will be the first major step in making a new life for yourself. If your ring is very modest and you don't expect to get much for it use the money for a little self-pampering. Maybe the money will only last for the weekend but make sure it's used for something that you will truly enjoy.
Give to your son or daughter.
If you have children you may want to pass the ring along to them. They might feel a little uncomfortable accepting a ring that's from a divorced marriage but assure them that when you first received the ring it was given out of love. You don't want them to think that they have to use this ring at their own wedding either. It simply can just be a family heirloom that's being passed down.
If you don't want to see the ring anymore, and don't want it to remain in the family either, another option is to donate it. This can be a great tax write off. This also ensures that the ring will truly be used to help someone else out. When you donate it the recipient may choose to sell it at a jewelry store, sell it through a thrift store, or the ring might be given to a couple truly in need of a ring for their own wedding. Wherever the ring may end up should benefit someone else greatly thanks to you.