Welcome to Benghazi
Depending on your budget the neighborhoods you'll probably want to look at are al Fwehat, al Hada-ek, and Garyounis, which are all (by Libyan standards) nice and quiet neighborhoods.
The first two are next to each other and are where most of the few Westerners here live. Garyounis is located on the coastline at the southern edge of the city. Just north of Garyounis is an area called "Taballino" which you might also wanna check out. But it's super expensive because that's where most of the rich Libyans live.