In System Settings -> User Accounts, you can tick on and off the Show my login name in in the menu bar option.
The problem being is that when I tick it off (I don't want my login name to be displayed in the menu bar), it also hides the Ubuntu cogwheel thing that lets you do things like logging in as a different user, rebooting and shutting down your computer et cetera. And I don't want that. I just want to hide the user name.
Try with Unity Tweak tool, to see if it'll work.
It's available through Ubuntu software or sudo apt-get install unity-tweak-tool.
In the application go to Launcher / Panel / Show my name.