I have a Toshiba Tecra 8100 operating on Windows XT. When I put it into standby mode, then bring it out of standby, the external mouse (Microsoft) is inoperative, but the little joystick mouse on the keyboard works. Is there a fix for this - other than not using standby mode?

Verify you've got the latest BIOS and USB Controller updates installed for your computer.
If these drivers are already fully updated, it comes down to hardware conflict
Try a different USB port, or try another type of mouse
There is nothing else to do, (other than re-connecting the mouse each time) many users report this same issue, on many systems

That the mouse is a serial device. How does one go about doing a BIOS update?

Actually most of these reports of mouse not responding is usually on USB

Serial mouse, is now very old, and not very plug and play friendly.
Forget the bios (for the moment) ie it isn't faulty, don't touch it.

But, yes replace your mouse with a newer USB mouse. (ideally suited for Laptop)

try the hibernate option instead of standby; it saves power and performs a fast reboot
into your desktop.

I'm w/USB mouse and found hibernate to run auto-discovery coming back running

prglor, update BIOS, and ESPECIALLY update the EC/KBC firmware. the serial mouse works via KBC controller and there can be mistakes in firmware.