Have you checked the port pins to see if any are bent/missing? It does happen...
According to the schematic I have, the port(s) run the traces through EMI filters. One or two of these may be shorting or open. It's hard to describe in full detail what these things do, but you'll see three pins: IN, OUT, and Shield. The in and out is actually a feed-thru, and the Shield is a capacitive/ferritized thingy that gets rid of extraneous EMI (like the big ferrite beads you see on some mobos/cables). The in and out are connected, so should read as a short (continuity good), and the shield should be isolated from the pass-thru (infinite or open). If you get a reading between the shield and either in or out, then it's shorted. If you don't have continuity between in and out, then it's open. You can remove and just use a jumper from in to out in a pinch.
After that's checked, follow the leads to a 74LS155 IC (U34), and check it with a logic probe. You should read pulses on pins 2, 3, 4, and 5 as you move the mouse around. Pin 13 should pulse an output based on the mouse inputs. There's also another EMI filter (E374) to check after pin 13.
After that, you're looking at pin 20 of LISA (the MDAT line runs to Lisa for mouse data, the ODD_CIA reads the fire buttons, and Paula reads the pots). Also check the pull-up resistor pack (R370) to make sure they're all good (4.7K, or 4K7 for our overseas folks).
This is all based on your claim that the mouse works with another Amiga. The most common problems are a bad LED/diode inside the mouse itself, followed by a bad 7414 chip (Schmitt buffer for the V/H quadrature), a blown fuse for the mouse's 5V, and then the above list. To check the LEDs, just point a remote control at various digital cameras you may have, and see if they can "see" the IR signal. When you find one, just pop open the mouse, plug it in, and you should see the V/H LEDs light up on the camera's screen.
banzai