Like the title says maybe you could add a set bonus that gives an influence bonus on kill with the amount gained per kill increasing with 3, 4 and 6 set bonus items. The amount should be small so not as to make it overpowered, something like +1, +2 and +4 per kill or maybe vary it for different strength npc's.

I've always wondered why this isn't a thing. Companions should grow to like and trust you more (increased influence) as you fight along side them more. Every time you kill a mob, you get a bit of influence along these lines:

 

Weak: none

Standard: 1

Strong: 2

Elite: 3

Champion: 4

 

Influence totals are so high now that it's impractical to farm. It takes like 250,000 total points for rank 50? Getting a few points for killing mobs would be a great touch to improve Companion management.

I think this would be a great modification to the companions influence gain! More then most days, the main way in getting their influence up is through conversations but when the story is finished and the companions isn't complete, you're left with gifts to finish it off, and gifts can be expensive. It'd be nice to have more control on the companions influence gain. Great Idea!

I would support this.