I have no specific experience with the Macintosh Portable, but generally speaking such a problem could be due to one of many different types of faults:
-internal fault in the LCD panel itself
-fault in a driver IC (on back of LCD panel PCB)
-fault in the computer's video memory
The easiest way of ruling out the later, if you can get another LCD module to swap in. That will tell you if the fault is in the LCD display module or the computer (video memory).
Since the line does not go all the way across the LCD panel a bad connection to the LCD panel glass edge can be ruled out.