According to this, it seems that you can use Java in Firefox extensions:
https://developer.mozilla.org/en/java_in_firefox_extensionsSo yeah, if you modified the source code, you could probably do something like that, and since it wouldn't be an applet anymore, you could probably add an option to have it only kick in if the screensaver is on, or if there hasn't been any mouse/keyboard input for X number of minutes, so it would only run while the computer isn't being used.
Actually, if you can run non-applet Java code in an extension, it kinda makes me wonder if it'd be possible to get a Java gpu miner running in an extension too. The main problem with that would be that the computer would have to have a graphics card capable of using opencl, and the appropriate sdk would have to be installed.