Interesting idea - Do the NPCs already have pre-existing relationships before the game begins? Bob and Joe are friends, Charlie doesn't like Joe, everybody hates the manager?
That's what I was thinking, but something about that may have to be randomized so that it's not the same puzzle every time.
What about having to deal with the emotions of distraught NPCs immediately after a disaster? What comes to mind is the Chilean Mining Disaster a few years back, where supposedly the mental health of the miners, as led by the mining foreman, played a important part of their survival.
Great idea! Maybe each level is a different scenario? New guys come in, Charlie's girlfriend left him for the company pencil pusher, etc.

A construction site would also offer many good ways of getting rid of the bad blood 
Chris Crawford's game Balance of Power would end with a black screen if you brought the world to nuclear war, and some text would say, "No, there is no animated mushroom cloud with parts of bodies everywhere. We do not reward failure." So, if your team ends up killing each other, I think it's game over.
The first thought I have is that, for this to be a "game" and not an educational simulation, you need a primary mechanic and a goal. If we run with the construction site thing, that gives us the goal - complete the building. It also gives us subgoals in the various jobs that need to be done, which is great.
The good news is that my goal is not to make a construction site simulator. I wouldn't even be the right person to do that. The goal is to have the player lead a team to accomplish some visible task, and efficiently mete out some problems in the group.
Primary mechanic? Organizing your construction teams for maximum efficiency. Certain people don't work well around other certain people.
And talking to them, too.
Sometimes they rub each other the wrong way. Sometimes pairing up two slackers encourages them to slack off even more. Sometimes their combined experience isn't enough for the task at hand, etc etc etc.
The trick is to balance the information you give the player with the information you withold, I think. Unless you want to completely simulate human emotional behavior, you need to heavily abstract it. That might lead to altogether too much predictabilty, though, unless you get that initial balance correct. My thoughts? "Professional" information should be given: Skills with various tools and tasks, experience and work ethic, etc. "Personal" information is hidden though, and heavily affects how a person works within a team. Thus the job of the player (manager) is to suss out enough of a person's hidden personal characteristics to put them in the best group possible, and THEN balance the combinations of people and groups to make an effective set of teams.
Interesting. But I wonder if the construction goal will start to override the goal of managing people. Play-testing is a must. I'm not interested in modeling the human psyche, either, so there will be a lot of abstraction to keep things simple.
At least, that's the first idea I came up with. Modeling that personal behavior in ways that is at once believable and unpredictable (more problems come out under the stress of close deadlines, for instance, so problems that might not have shown up before suddenly appear) is the meat of the game and needs to be nailed.
I agree. Thank you!

I really like the idea of a construction site! Modern settings have a special place in my heart. 
I was really worried about that when I posted this.

I don't want fantasy, I want to do something a lot more grounded, so it's great to see some interest there!
Like Telcontar said, I feel that having the player discover or some how figure out the workers traits would be a good way of managing them, even if it is just trying to glean those traits based off their interactions. Having each worker have something like a name card that you can check off what traits you think they have. Like Joe could be moody and Max is an extrovert, making the chance of them fighting a bit higher than if either was paired with the introvert slacker Mike.
Then, having their traits unknown at first but giving a list of possible traits for each worker will help the player feel less lost in the process and once they master the system, they will know exactly why. It'll give them confidence and satisfaction from playing the game, which is pretty dang cool.
If the workers have random personality traits that work well with some while completely clashing with others, you probably wouldn't need to have pre-determined relationships since those relationships will appear based on how their traits mesh. An angry, alpha workaholic being paired with 2 passive workaholics could get along great but 2 alphas together and you have a mess to clean up.
I think that makes a lot of sense.

I'd imagine this information can be play-tested a little bit as a kind of card game or board game, too. Not really any construction, of course, but you could have players role-play a little bit. Each player who plays a worker takes a "skill card" and also a "personality card". The final player is the boss, the player in the computer game, and he gets a copy of each player's "skill card" and goes about managing his team. Maybe have a board to show where each player is on the site, as well.
My only other thought is finding ways to force the workers to work together. If there are enough things to work on, then they could just be separated and won't need to interact. If you make it so some tasks can't be completed without 3 or more workers and tasks are dependent on one another, then you have a pretty simple system that has a high chance of chaos going that I think would be a really fun resource management game.
Put the pressure on to get the results you want.

Just my thoughts, but now get to work! I wanna play it. 
Right on!