You may have access to the system tray in an applet (I'm not sure). Have a look at the java.awt.SystemTray class - the in-tray lets you pop up messages to the user.
Alternatively you could attempt to cause the Window's toFront method to be called or to "maximize" using the setSize methods (again, I'm not sure what effect this has in an applet). I suspect that the toFront method will be a good bet
Another option I'd look at is raising a JDialog. The presence of this may cause the OS to draw attention to the minimized applet. You could listen to window events representing the screen un-minimizing to clear the dialog so that the user never knew it was there.