I would like to give a user the ability to clear the messages from the Text
Chat area (and it would be nice if they could save the old messages too). I
have found how to do it for them, but my method requires restarting the
application.

You could do a set property of the the shared object data element that holds
the text chat string to null or ""

To allow the user to save the messages, you could save them to t different
data element of the shared object on the server or perhaps write them to a
local shared object onthe user's machine.