If it's just a basic assignment and all you have to do is get that users name to be displayed until they choose to log out, it's really quite easy. I can't do it for you, because it's your homework, but I can nudge you in the right direction pretty far.

session_start()
$_SESSION superglobal

That should be all you need, along with $_POST and one form to ask them their name. One if statement and your logout can be taken care of in one method call too.
-m