Could be a number of things... The Lisa mouse is a quadrature mouse (basically the same as a "bus mouse") and so there are two signal lines each for horizontal and vertical motion. (
See pinout here.) If one of the two lines for vertical motion isn't working, then you might see something like what you see here.
The fact that it bounces up at the top of the screen suggests to me that the problem probably isn't in the mouse itself, but if you wanted to confirm that, you could hack up a little "adaptor" cable that swaps the X pins with the Y pins. If you see the same behaviour (albeit with up/down motions of the mouse now making the mouse pointer move right and left), then it's not the mouse.
So once you have localised the problem to the inside of the computer, then you've got a few potential culprits. If you've had a battery leak in your Lisa, then the card edge connector on the motherboard that receives the I/O board (the closest board to the back of the computer) could be corroded or hopefully just dirty.
And if it's not that, then there's some pullups and a 74LS153 (see location D2 on
schematic page 2) between the mouse port and the COPS, the Lisa's microcontroller for keyboard/mouse/power button/RTC. If the COPS were broken then a lot of other things wouldn't work, so I suspect that the problem lies somewhere between the COPS and the mouse port on the back of the Lisa.
See also PDF page 223 of this copy of the
Lisa Hardware Manual.