When I first opened my Facebook account I wasn't really sure if I would like being a Facebook user so I opened my account with my maiden name which was Carol Baker, not thinking I would still be on 3 years plus later. The problem is my husband's family (who live across the other side of the country) don't really understand why someone with a surname different from theirs wants to connect with them. (My husband has explained but now it seems older members of his family are wondering if we have divorced without letting them know. They think "why would a married woman want to use her maiden name if she is married to a person with a different surname?") So it would be for the best if I were able to use my real name: Carol Smiles. I have tried but still Facebook won't allow me to. Do I really need to close my account? I would be hesitant to start all over again. Besides Facebook does say "real names must appear on all accounts".
On the top right corner you can see a star/gear looking icon. Click on it and it brings a popup menu. Choose Account Settings which brings up General Account Settings page. First line is your name and to the right you have an Edit button. Use it to change your name.
Facebook doesn't believe your last name is really "Smiles". "Smiles" appears on a list of names that are not "allowed" by default.
Follow this to provide proof to Facebook with details of your real name, and they will help you resolve the issue.