How to increase influence in Stellaris?
Original post:
Tech. There are several social techs providing +1 influence/month. Additionally one of thsoe provides an empire unique building with +1 / month.
Declare rivalries. You can have three rivalries, together they'll grant 2-3 influence per month.
Complete the mandate. If you play a democratic empire (elections every few years), your elected ruler will declare a mandate (e.g. build 4 mining stations), which will grant a significant influence boost (lump sum of 80-200)
Reduce expenses. Empire Edicts and frontier outposts cost 1 influence per month. There is also tech to reduce this cost.
Some anomalies and events may also grant influence.
Update 2019:
Since release the ways of generating influence significantly changed. The wiki provides a comprehensive list. Additionally, the factions in your empire play a major part in generating influence for you.
Addendum to the old points:
Techs are severly reduced. Autonomous agents (for gestalt conciousness) grants +1 influence and living state (for all other races) grants +1 max influence from factions. Interstellar campaigns and galactic campaigns reduce the influence cost of claims. That's all influence related tech left. The empire capital complex building also no longer exists.
Rivalries are still there, but only grant 0.5 influence per rivalry.
Mandates are still in the game, but now grant Unity, not influence.
Empire edicts no longer cost influence per month, but a lump sum to activate for a 10 year period (baseline), outposts are now needed much more frequently than in the release version, they also cost a lump sum of influence based on the distance to your territory and have no influence upkeep.
remains unchanged
One thing no one else has mentioned is at least in the updated versions you can gain influence by getting factions over 60% happiness
I have found an additional way:
The building "Empire Capital-Complex" that can only be built once provides +1 influence.
But! If you conquer a planet which already has one, it doesn't get destroyed, and you also get +1 influence from it.
So in short; conquer a planet with an "Empire Capital-Complex" on it.
I've found these to mostly occur on capital worlds.