How do I replace the MeMenu username with my actual name?

To switch to your actual name enter the following in a console (Applications->Accessories->Terminal) or you can use the gconf-editor:

gconftool -s /system/indicator/me/display --type int 2

And to switch back:

gconftool -s /system/indicator/me/display --type int 1

It is also possible to hide the name altogether and show just the icon:

gconftool -s /system/indicator/me/display --type int 0