It looks to me that the text of the rule is saying that you CAN "activate" it multiple times per use.
p281 - Core Rulebook:
Force Power: Influence
The most basic form of Influence does not allow the Force user to guide or shape the thoughts of others. He can "merely" strain their mind, inflicting stress and exhaustion. The basic power has one effect that can be triggered multiple times on the same or different targets. (bolding and Italics mine)
- The user spends one light side force pip to stress the mind of one living target he is engaged with, inflicting 1 strain.
As I read it, the issue is the lack of clarity in the Strength upgrade wording in the chart...it doesn't explicitly say two per pip...it only says "When stressing the mind of a target, the character inflicts 2 strain."
I read that as two per pip, not one since the text box for the upgrade (p283) does say:
Strength Upgrade: When stressing the mind of a target, the Force user inflicts 2 strain instead of 1.
This means that a Force Sensitive individual with even a single Force die COULD be great at dealing with minions, overwhelming their minds bit by bit (i.e. killing them since they don't have strain, I guess) if he's lucky with his rolls.