Click on the account, select the account setting.
When you are in Account Setting page, click on "Change" (same row with Name)
Then come to the page that you could change your name.
First Name & Last Name column are mandatory, while Middle Name, Full Alternative Name are optional.
Click on Change Name when you are complete.
You have to 'tick' on the check box ("i confirm this is a ....."), then click on confirm request botton.
(you may double check your display name before confirm your request)
Not every request will approve by FB, please take note on this.
1. FB take AT LEAST 1 day to approve your request.
2. FB will automatically filter out some names that it think not suitable, and reject immediately, so, keep trying different name until you get approve.
3. User have the option to display or NOT to display alternative name.
Hope this help.