your name on facebopok profile which you take at the time of registration. You need to send a request of name change to facebook. Here are step by step guide for how to change your user Name in facebook.
How to Change Name in Facebook account?
1. Login to your facebook acoount.
2. Click on the “Account” Button at the right of the page and select “Account Setting”.
2. Click on the “Account” Button at the right of the page and select “Account Setting”.
3. Now find Name section and click on “Change” link at the right
4. Fill out the form with your first, Middle, and last name. And click on the save botton.
You down your part. Now facebook review all the requested one by one , and may be take upto 24 hours to change your Name. There is a limit on how many times you can change your Name. so be careful.
I think you are able to change your facebook name, but if you have any problem please tell me via comment. I try my best to solve your problem.