There's a reference to the respect stat in the latest changelog. Does it actually do anything now besides making your wives less jealous towards each other? And is there some new way to raise it, since you'd only get 1 point of it very rarely before. Unless you have them join you in combat, which many aren't very good at.
Bonus to Academy rolls, meaning they have a slightly better chance at learning, up to caps. It's a check for bringing people as followers in the rebellion I think. It might affect their willingness to go get surgery, for mercs. It's also a bonus to some persuasion checks I imagine.
How good they are does not matter when they can have stealth armor and you are clearing trash bandit mobs.
There is a passive daily chance to gain respect up to 50 if you have the Defender of Ikaanos trait. For that, you need to defeat the Dengi King's group in the Dengi invasion scenario set in either character creation or game settings page 3. The number indicated is the number of days until the invasion can commence. Do not start it unless you are extremely ready. The Dengi King is basically Terror from the Deep. His escorts are a sorcerer mutant (diet Mordecai), two beasts equivalent to Feral Charucks, and an equivalent to Palace Warmaster. Approximately.