I want to advance in the ranks of Senator Lavigny-Duval. How can I get merit points? I know I can participate in expansion goals, but is there a better way than that?
You get merit points by participating in pretty much anything Powerplay related. Expansion, Fortify, Prep, Undermine, etc.
The easiest and most painless way to do it is by Fortifying (although it is very time consuming). You do this by collecting fortification materials from HQ and delivering them to a control system that you own. Preparation is a similar action, just taking the data to a new uncontrolled system from any controlled system. (1 merit per item)
Expansion for Arissa has you enter the prepared system and enter a conflict zone like area. Other powers have different objective, but it's still the same rewards. The merits aren't much (1 per objective) but a decent amount of time and hanging around allies could get reasonable merits pretty quickly.
Undermining and Opposition is by far the best way to get merits, but it is also the most dangerous. To undermine powers not aligned to your major faction, you must go into their controlled system and kill specific transport ships, either by interdicting them or hanging out in the nav beacon. Undermining and opposing an ally major faction has you pirating these transport ships instead of killing them, try and do this as non-combat as possible, they are allies after all. Be warned, this will be an illegal act and attract bounties on your head as well as the security force coming down on you. (15 merit per kill/loot)
If you attack any ship that is in a power aligned to your major faction, you will lose merits, so always avoid lethal engagements with allies. (-10 merit per ally kill))