As much as I really liked the idea of completely procedurally generated empires, the more I've played the game the more I've come to dislike every empire having access to every trait, ethic, and civic. While I don't want a fixed roster of Empires in every game, I would like to be able to make some predictions of how types of species will behave.
One fairly simple thing to do is tie species type to habitation preferences. For example, of favorite starfish might like ocean worlds, the space kitties could prefer savannahs, and the cockroaches can be found at home on tomb worlds. Note that I wouldn't any of these to be hard and fast rules, just weighted probabilities to make certain outcomes more likely.
Slightly more complicated is making certain ethics or civics more likely for certain species. A species with large, predatory fangs might be more likely to be militaristic, while (fuzzy butterflies)<https://stellaris.paradoxwikis.com/File:Arthropoid_18.png> would be less likely. This also could extend to species type generally–mammalians might be more likely to be xenophile, or bugs might be more likely to be collectivist. Some of these decisions could be pretty arbitrary, but I personally think that's fine.
Along the same vein, we could make certain species more likely to have certain traits–birds more likely to be weak, Molluscoids more likely to live longer, etc.–or restrict certain traits to specific species. For example, we might make borrow some traits from a particular extended trait mod and say that only plants can by photosynthetic, only these guys can be parasitic, the venus fly trap can't be an herbivore, etc.
I have no idea if this would actually work out, and like I said a lot of the decisions could end up being arbitrary, but I do think imposing a little order onto the currently completely random empire generator would be a good thing.