This is my second post on this thread, but it sounds like everybody's got a different opinion about what you should do. Don't over think this. You tried to locate the owner of the ring when you found it and you've kept it for 10 years. Your post asking opinions says to me that you could be hesitant to sell it or you could just want someone to say it's okay to do it. At the end of the day, whatever you do will be on your conscience, not those of the people who respond.
Some years ago, a student found a 1980something class ring on a baseball field at another high school and brought to my school. He asked that I try to find the owner, but didn't want to leave his name. I convinced him to leave it anyway along with his phone number. If the owner could be found he would at least want to thank this boy for returning it.
To make a long story short, it was pretty easy to id the owner using the initials inside the ring, the year of graduation, and the commencement list from that year. All the records had been sent to the central office, but I found the name in the phone book. The boy who lost the ring luckily was a junior. Surprisingly, the student's parents still lived in the same house and had the same phone number they had when their son was in school here! I called and left a message. The next day, the guy who had lost the ring called me back. I told him about the kid who had found his ring, especially the part about not wanting to leave his name. He was so happy when he picked up the ring he'd lost 15 or so years earlier! He left an envelope with a cash reward for the kid who found it, along with a note commending him on his honesty.
It will be definitely be harder to figure out the owner using a ss number since they are kept confidential. You could call the school and ask them first if they keep these numbers on file anywhere. If the answer is no, then I'm not sure where you'd go from there. But if the answer is yes, you could ask them to call the person it belongs to an have that person call you. They will not give you his number or address or even his name due to confidentiality laws, but should certainly be willing to do ask you ask. One thing I can say is if the ring is more than 20 years old, the school will probably have to contact their central records department, the same way you'd search for a transcript.