Technically the lowest amount of damage possible is 0, if you use Hold Back or False Swipe on a Pokemon with 1 HP remaining.
If the target has 1 HP remaining, False Swipe will hit and leave the target at 1 HP. - Bulbapedia
Excluding this however, the lowest amount of damage is going to be 1. There are so many possible ways this could happens that it'd be impossible to locate each and every one of them.
If you are asking a more debatable kind of question, like what would be the weakest attack realistically and not mathematically, it really depends on the person and what they believe is weak.