October 2017
It would be nice if we had missions that we could do daily to raise a companion's influence. For instance, examining Sith artifacts with Lana, helping Theron sift through reports, brainstorming for experiments that those of our ethical Commanders may have to put limits on with Oggurobb, and so forth.
It seems like there should be some kind of social mission tailored to each companion that we should be able to engage in. For instance, missions like C2's comfort mission but have it raise the companions influence instead of other NPCs.
March 2018
I apologize for the delay in a response. If you haven't already I would also recommend posting this in the official Suggestion forums found here for the best visibility and to add more discussion to your suggestion for other players to see.
March 2018 - last edited March 2018
Gifts are too heavy for raise influence, we need more talking, visit their homeworld, romance or something else to do.