If you are asking me what work would need to be done at this stage in the development cycle it is impossible to say without know exactly what you have done, whether it would be conducive to such a system and whether such a system would compliment what you already have done and vice versa.
No, I'm asking you to give some thought as to what elements this feature would actually entail. What level of detail are you looking for, and what work do you think would be required downstream.
Assume you have no restraints on your budget as at this point we're just brainstorming.
Your OP is a bit sparse aside from "let them do more than just sit around waiting."
But I'll be a bit less difficult and hopefully get the ball rolling: Conceptualizing and creating the plot arcs for companions while not in the active party
What might this entail?
- Creating a narrative. Is it cohesive or just one off things for flavour?
- Will it require spoken lines?
- Are there tangible in game consequences for any of these narratives? If so, is there any branching factor?
- Are we creating these on top of any interactions and quests that may be character specific with said NPCs that are actually in your party?
I understand that people will probably miss stuff (I will probably miss stuff), but lets detail out the specifications of the request because it's entirely possible that what I think satisfies your request is actually no where near sufficient for what you would like it to be.