Interesting project... I'm not sure of a quick way to program a serial driver for the IIe's game port. I was always more of a Commodore 64 guy myself (and it did have an RS-232 serial port, so if you've got one of those kicking around it might make your job easier... I could even scan the relevant pages from the manual and send them along).
But I do know that a cheap PIC chip can be programmed using the "serout" command of PIC Basic Pro (
www.melabs.com) to do precisely what you want. Well, what you want from a technical standpoint, that is. There is a certain aesthetic statement in using the IIe, I suspect.
Doing a Google search for RS232 should help you to find documentation on the standards that drive typical PC serial ports, but you'll have to program your own timing loops to send them.
Good luck,
Jason