Edit: TL;DR included at bottom.
I'm also interested to know the specifics of exactly what gives authority, so I'll make a detailed list. A lot of these traits have triggers where you have a chance of getting them at birth, at adoption, or at marriage. I'm not going to include those for each and every one since you can't do much about those anyway. Also, I'm only including the traits that give out authority at their highest level. There are a couple traits that give authority at lower levels, but then give either negative or no authority at higher levels. I'm not including those traits.
Also, being a faction leader gives you 2 extra authority. Being a faction heir gives you 1 extra authority.
Disciplinarian gives up to 1 authority.
So it appears a FM can get it if a settlement they are governing riots or rebels.
Brave gives up to 2 authority.
So your FM can get it if he loses a bunch of health in battle, gets a bunch of kills in battle, or if his father has the coward trait.
Energetic gives up to 2 authority.
Your FM can get it if his father is either slothful or energetic, though it could go either way so you can't really rely on that. If he already has the energetic trait and has less than 10% action points remaining at the end of a turn, he has a chance to gain more energetic points. So basically you just kind of have to hope that the FM gets it at birth/adoption.
PoliticsSkill gives up to 2 authority.
Anytime you do diplomacy or bribery it gives the chance of your FL getting it.
Rabblerouser gives up to 2 authority.
So a FM can get it when a city he is governing rebels or riots.
VictorVirtue gives up to 3 authority.
A FM can get it after winning a siege battle with a clear victory, he got at least 6 kills, and the odds of the battle were not in your favor.
DeceiverVirtue gives up to 3 authority.
A father with the Politics Skill trait can give this to their son. Also, ordering an assassination, spy mission, bribery, or sabotage has a chance of giving this to your FL.
Authoritarian gives up to 2 authority.
A FM can get it when a settlement he is governing riots or rebels. He can get it when one of the buildings in the town hall line is completed when he is in the settlement. He can also get it for exterinating a population after conquering a settlement.
BattleScarred gives up to 2 authority.
A FM can get it if he loses enough health in a battle.
Handsome gives up to 1 authority.
A FM is more likely to get it if he has a handsome dad.
Pragmatic gives up to 3 authority.
A FM can get it if his father is pragmatic. He can also get it if he ends his turn in enemy lands outside of a settlement.
Prim gives up to 2 authority.
A FM can get it if he ends his turn in a town with a religious building, if his dad has either the gay or likes girls traits, if his dad has the cuckold trait, or if his dad has the prim trait. A FM also has a chance of getting Prim at marriage or when they have a child.
BattleDread gives up to 2 authority.
If your FM gets a bunch of kills and he doesn't have battle chivarly, he has a chance of getting battle dread. Also, winning battles against weaker foes where you kill most of them (as opposed to letting them rout) gives a chance for battle dread. Also, winning a crushing victory with slightly lower odds with more than 70% of your enemies killed gives a chance for battle dread.
BattleChivalry gives up to 2 authority.
If your FM wins a battle and gets a bunch of kills and doesn't have battle dread, he can get battle chivalry. If he wins a battle with low odds he can get it. A Catholic or Orthodox general assaulting a Muslim general can give battle chivalry. A Muslim general assaulting a Catholic general can give battle chivalry. Killing very few of your enemies while letting most of them rout can also give battle chivalry.
StrategyDread gives up to 3 authority.
A spy mission, assassination, or bribery can give this trait to your FL. A FM can get this trait by accepting a bribe. A FM in a settlement with high taxes, low happiness, and a treasury greater than 10000 can get this trait. When your treasury grows to more than 50000, any general without strategy chivalry can get stratey dread. The governor of a settlement that creates a spy or assassin can also get this trait. Finally, exterminating a population after conquering a settlement can grant this trait.
StrategyChivalry gives up to 3 authority.
Refusing a bribe can give this trait to a FM. A FM can get this if he's in a settlement with low taxes and high happiness. Having a negative treasury increases these chances since only a chivalrous governor will set low taxes when he has no money.
FactionKiller gives up to 4 authority.
Your FL can get this if he delivers the killing blow to a faction.
GoodDiplomacy gives up to 4 authority. There doesn't seem to be any triggers for this though. Only way to get this trait is through console command cheats.
FathersLegacy gives up to 3 authority.
A FL's son can get this if the FL has the faction killer trait.
TourneyKnight gives up to 2 authority.
A FM can get this if he ends his turn with 100% action points remaining in a settlement with a jousting building. He can also get it if his father has the tourney knight trait.
HorseRacer gives up to 1 authority.
A FM can get this if they end turn in a settlement with a racing track and with 100% action points remaining.
Cultured gives up to 1 authority.
A FM can get this if they end turn with 100% action points remaining in a settlement with an Ikoner's Studio, Printing Press, University, or Artist's Studio. They can also get it if they are in a settlement when it completes building a theater or artist's studio. A FM also has a chance to get the cultured trait if their father has it.
TL;DR Family Members can get Authority by generally taking actions that give them either very high dread or very high chivalry. Being neutral is no good. Governing a settlement when it rebels or riots can give them good authority traits. Taking part in a battle, losing some health, and racking up some kills can get them some authority. Being in a settlement when an artsy type building, town hall, racing track, or jousting arena is completed can give authority traits. Attacking a Muslim faction as a Christian or attacking a Catholic faction as a Muslim can give authority as well. Those are most of the basics.

Originally Posted by
PrestigeX
I'm not entirely sure but ... the things I've found that help my Faction Leaders / Fam members are.
1 - release prisoners, and occupy (Rather than ransom/execute, or sack/exterminate) - becuase this increases the chivalry, makes your faction more reliable, and generally being 'good guys' yields many positive attributes.
2- building the town hall lines of buildings - increasing law and order
3- Use Kings/Faction heirs in battles, esp in crusades, and also to rid the land of brigands and rebels.
4- Do not use the brothel line of buildings , or assasins etc .
IdK I might be totally wrong, but i tend to notice a correlation between Chivalry , Command, and Authority
You definitely seem to be right about 2. Number 3 applies to all family members. Not just FL and FH. Make sure you actually use them to get some kills and not just sit on the sidelines though. Number 4 and 1 aren't really true. You can get high authority with high dread or high chivalry. You don't need to go with high chivalry.