If you're not that familiar with the choices, I'm sure you'll find
PHP/MySQL much easier to learn and get running. On the other hand,
there is really no good way to update the client asynchronously with
that method, so the users would have to click something every now and
then to see new messages. You would get better performance and
usability from the JSP, etc, approach, and be able to update
asynchronously without reloading the page, but the learning curve for
those technologies is quite a bit higher. Probably don't really need
flash, and that's another whole learning curve.

Incidentally, there is a package called FlashChat that uses flash  on
the client end and PHP/MySQL (I think) on the server end. Works
rather well. Costs $5, but that buys you the source code, too, even
the Flash FLA file. Might be worth it if you want to see how it
works. You can see it operating on a prototype site I did at
http://clay.duane-n-lisa.net/kids/chat.php. (You have to register to
get to the chat  client, but you can put in anything you want.) I'll
try to look up the link to the author if you're interested.