Gifting in Stellaris to increase opinion

According to this wiki, the save game files are located here,

%USERPROFILE%\Documents\Paradox Interactive\Stellaris\save games\$EMPIRENAME+ID\

It does not seem to indicate any difference for different game modes, or if cloud synch is enabled, however the save game files are zipped.

The savegames are zipped. In order to edit them, you'll need to extract them. There are two text files: gamestate and meta. meta includes the information that is shown on the load game screen. gamestate is the actual save game.

Original post:

1. Tech. There are several social techs providing +1 influence/month. Additionally one of thsoe provides an empire unique building with +1 / month.

2. Declare rivalries. You can have three rivalries, together they'll grant 2-3 influence per month.

3. 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)

4. Reduce expenses. Empire Edicts and frontier outposts cost 1 influence per month. There is also tech to reduce this cost.

5. 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:

1. 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.

2. Rivalries are still there, but only grant 0.5 influence per rivalry.

3. Mandates are still in the game, but now grant Unity, not influence.

4. 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.

5. remains unchanged