What is the purpose of players attempting to change a faction's influence?
There is a large element of roleplaying and fun to the reasoning behind players supporting certain minor factions, but there is also a gameplay element that has to do with PowerPlay.
When dealing with a system that you power wants to expand into, control, fortify, etc. It's much better if it has a government that matches the power's ideal, as in it cuts the requirement needed.
The same tactic can be used to undermine powers, by changing the government to the one a power does not get along with, the cost of upkeep is increased.