Well, my thoughts were the ones of a classic resource/time strategy sim, where you have to scavenge for "materials" (up to a discussion for what kind of materials, depending on the scenario), use them to raise the camp/hideout so they boost some of your stats (that will go up anyway while using them with the scripted actions).
As for what concerns the actions i thought about some of the classic survivalist actions and added some with a different flavour (notice: what follows is just some of the ideas, i just wanted to throw them around before they'd go lost in the air xD):
- Scavenge for [Material name] with a chance to find nothing, regular, double OR find someone you can decide what to do with them (that would decide if you're going full villain with an actual "kingdom" of slaves of if you want to keep your family safe assuming you have one)
- Look for survivors same as above just with a slighter more % of finding someone and a 0% of finding any kind of resource
- A chance to open "The Pit" where u can have your slaves fight (up to you to decide if to the death or just as a competition) and raise their submissiveness or aggressivity
- A chance (dunno if possible, but considering the actual house party uses up to 6 ppl it should be doable) to add more than 1 person as companion and singularly be able to give them orders, that would be amazing to add random encounters with zombies/other groups that might end up with a fight where companions are the easier strenght you can find. Also pairing with companions could raise a "trust" variable for how much that companion would put his/her life at stake to protect you and so on
- Specializations for weapons. When u have resources u can make "artisans" show up at your gates and they can create different kind of weapons. The more u use one the better u get at using them. Im not talking about adding RPG style encounters with HPs and stuff like that, im talking about the chance to defeat someone with one, two or three hits or something like that so it's like a % damage on the total HPs of the encounter (that could go along with the previous point, considering how many companions are following you u will hit more than once in a "turn")
...i could go on for a lot more, i just wanna find out if the ideas i'm throwing are going along with yours xD