OK, forget the bonuses part, they grant nothing 
I was thinking maybe call them based on lineage? Like noble descendants of warriors or caste of bureaucrats.
Or something more abstract like "Order of th White Swan"
I'm sorry, but what you're asking for is not really feasible.
Your asking for "cool" names for disparate factions, that don't have any back story, bonus, or purpose.
Your asking for a name before you have a theme or the product.
That's like me going to an engineer and telling him to make a "Zooban", he asks me what that is, and I tell him it doesn't matter what it is, just so it will look, sound, play, and work like a "Zooban"
Figure out the purpose of each faction 1st before you start naming them.
Names come after a function/ purpose/ product has been defined.
If you want them to be family lineages, then start with what each family's values are, who these people are, and then find/ come up with a name that subtly or directly relates to or has contrast to their values or history.
Look at your world, setting and time period, what names would be fitting?
If these factions vying for the throne aren't actually doing so, and you want some cool sounding factions that don't have any purpose or story, nor mechanic tied to them, why make them in the 1st place?
If they have no purpose other than to grant a superficial sense of conflict, then you might as well just call them A, B, C, 1,2, 3....
Your making more work for yourself by adding in these factions that seem to represent nothing nore provide any kind of story or game mechanics.
Take the now well known, but books original work: A song of Ice and fire = Game of Thrones.
Each family has values, crests personalities, etc. and for most, their names reinforce and symbolize their essence.
Starks are the winter harsh cold people, the word "stark" has a similar meaning.
Same for each of the major houses, each name tells you something about their lineage, and how proud they are or aren't to hold that name.
If your using the standard aristocratic court names, and say have "The Economists gain power or prominence, how will that effect the economy? If it doesn't in any way, then they really serve no purpose, if they don't have an agenda, then why are they there?
Every detail has the potential to be a design decision.
For example, a while ago on this forum, a person representing a project which the main player character was meant to suffer from Schizophrenia, and the game was basicaly kill the guards to escape the institution.
They went on and on about how they wanted to get the mental health thing right, and get the player to feel it, then I checked out their indidb page, and the game was all in 3rd person.
PS.
I mean no disrespect, but in the short time of reading some of your threads, I'm getting the impression your outsourcing brain storming and writing department tasks to this forum.
It's fine to do so, if that's how you want to do it, but if you directly use any of our contributions you'll need to source us in your credits.
And if you think we should be grateful that our ideas are making it into your game, I for one am cool with it, but I'm not chomping at the bit, so grateful that I let proper credit fly out the window.
Ideally, a forum should help you get your ideas out, and allow them to be critically reviewed, for feedback, and inspiration, you yourself should be coming up with these names and concepts at the end of the day, only you and (your team, if you have a team) should be the ones that know your game so well to know what will work and what will not in your context.
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