Originally, this was going to be added to another post, but I believe it stands alone.


Influence is political power, plain and simple.  Thus, influence should be gained using political science.  Influence is generated with Absolute and Relative Advantage


1)  Absolute:  The more attention a system gets, the more influence it generates.  Is this your homeworld (or the homeworld of a conquered foe) it comes with a bonus.  If you lose your homeworld, you function with a penalty until it can be retaken.  Do you control Auriga? A system with an Endless relic?  Focusing one's expansion for "quality" systems could provide a bonus in influence, also providing an improved reason to contest ownership of irreplaceable, unique elements of the game.


2)  Relative:  Is your system or Empire the best?  Does it produce the most F/I/D/S (Both as separate categories and combined) in the empire? the galaxy? Is it the center of trade for a sector or the entire grid?  Is it the most built-up?  Conversely, there can be penalties for empires with abandoned or neglected systems that fall behind in terms of development.  


This would add a new dimension to diplomacy in that war-like races may risk influence deficits vis-a-vis diplomacy-focused races in hopes that the spoils of war will prove a worthy investment for recalibrating relative advantage.  Thus, war can become a tool of diplomacy, a means by which advantages to influence and diplomacy can be contested militarily.