Janzour, Palm City, Regatta or Gergaresh
Acceptable, trailer trash, ghetto, ghetto^2
If you can get a house/villa (shared or otherwise) in Janzour then do that. It's got relatively cleaner air, and less noise and less traffic. Otherwise it's hard to name one area that's consistently "really good"
Use the search function and look in the FAQs thread for reviews of the other three.
I can tolerate most of what comes with the Libyan package, but one thing that always gets me was/is the noise: fire crackers, loud vehicles, weeding parties that start at 3am...etc
If your team shows you a place, don't just take it. Come back at different times and walk the neighborhood. Ask about your neighbors and whether they have kids and how old.
Look for subtleties that you may have taken for granted in the past: water & sewage leakage (even in new buildings); and functioning ACs, windows, doors and door locks.
You sound like you're in an unusual situation for an American in Libya, since there may not be any Americans or even English speaking non-Libyans at your work so be very skeptical but don't show it.
Let us know how it goes.