Handing in explo charts also works, along with the standard passenger missions.
To a hostile starport/EDO settlement smuggling prohibited stuff to the blackmarket will also help you in an Election.
Never tried the black market stuff but if the side I want to support controls a dock in the system with universal cartographics it is a quick & easy way for the well organised Cmdr to dump in large amounts of influence quickly.
Collecting explo data takes time of course, but unlike cargo doesn't have to be sold straight away and unlike bounties (which don't influence an election but contribute to war) can be sold bit by bit, allowing the Cmdr to only sell what's needed to win that day.
The usefulness of exploration data for faction support often makes it my go-to tool for any faction that has a dock and I collect it whenever I can to save for future tactical use.