To change your name on your YouTube account, you will need to join Google+.

To do this, log in to your Google or Gmail account, then click the Google Apps icon (9 dots resembling a keypad, just to the left of your avatar). Click on the Google+ icon. On the left of the page that appears, click "Join Google+." On the window that pops up, you can change your name, and this will change it across all Google products. You will need to select an option for you gender and enter your birthdate before clicking "Create Profile." Your name change will not update until you click "Create Profile."