OK, there are more then one way to do this, but I'll do it in two different ways.
dconf-editorOpen dconf-editor and navigate to apps/indicator-session and check the box behind show-real-name-on-panel.
Unity Tweak ToolLaunch Unity Tweak Tool and search for Panel section. There You can check the box before Show my name.
Now You can see Your (real) name in the far right on panel.