There is the base value of the star when you first find it. This can be zero, but sometimes a naturally high influence star pops up. Then population increases influence and so do social projects like embassies. Some wonders increase star influence as well as empire influence. Those are the concrete values I am sure of.
Next is the part I am not too sure of. I do not know if empire-wide influence boosters increase star influence directly or if they just amplify existing influence that has already been collected from the systems. These include +influence picks at the start of the game, influence resource starbases, technologies like artificial satellite, and wonders like Galactic Exhibition.
Then there is the odd-ball of the pack: culture starbase. This multiplies the combined influence already in a sector, but does not adjust system influence.