Step 1: Underneath your profile photo on your LinkedIn, your current URL is listed. If you hover over it with your mouse, an option to edit appears. Click this and you are led to a page where you can edit your public profile.
Step 2: On the right side of this new page you will see this box. Click on the pencil icon and insert your full name into the box. Voila! Easy, right? Don't forget to hit save.