Here are some more suggestions ! I took some inspiration from a thread that was around a few days ago.
Militarist Fallen Empire :
Endless war : Stuck in an unending conflict between different factions of the empire since centuries. Will eventually unite against a big threat, either the player or a end-game crisis. Factions of the empire will eventually racket a minor species for energy credits or minerals to finance the war effort (That's how this empire is "triggered". Notice that it is random but influenced by your empire's wealth).
Honorable warriors : Wait for the day an empire as mighty as theirs will rise up and fight them in an epic battle. Triggered if you become "worthy" aka create a fleet as huge as theirs. Sometimes "test" you just to see if you have a chance of becoming worthy.
Individualist Fallen Empire :
Counterproductive democracy : Stuck in a decadent democratic system. It's like, they want to do something, but then the senate disagrees with it, then they have to pass a new law, then a new senate is elected, then they forget what they wanted to do in the first place. Triggered if you do un-individualist things, like enslave people or be an autocracy. Always help rebel factions.
Greedy mercantilists : Rich and decadent as fuck. A single of their citizen is more wealthy than the richest of your oligarchs. Wants to do trade deals with other species. Sometimes they ask you for weird stuff, like a handful of your citizens as slaves for their golden palaces. You don't really want it, but they can pay you so well... and they might be really, really angry if you don't give them what they want.
Collectivist Fallen Empire :
Long-term builders : Obsessed by the construction of an enormous ship/building/statue/thing. You don't really know what they're doing, but it's huge and they've been busy doing it for centuries. They need a lot of rare ressources for this thing. If you appear to be in possession of said rare materials, they might decide that you're not really worthy of this thing and take it away from you.
Perfect society : Confident that their society is perfect and cannot be updated. Brag about it constantly. Triggered if you decide to change your base society to an "advanced" society, because they don't want any rivalry.