The valley is called Gei
Ben Hinom - Velley of the Son of Hinom, if you want to be
literal. Even if you don't,
gei is definitely the
possesive conjugation of
gai, meaning valley, and not an
integral part of the name.
The closest gate to the valley is the Jaffa Gate, not the Damascus Gate. The valley runs along the western periphery of the Old City, on the ancient road to the sea (and hence Jaffa).
The first Jewish neighbourhoods outside the confines of the Old City were built on the eastern slopes of this valley, and it was for many years a lower class area inundated with crime and poverty. Nowadays the more quaint locations overrun with American millionaires and successful artists with wealthy backers - no normal person can afford to live there anymore. A large portion of the beautiful old building has been torn down to make room for the new David's Village (or something to that effect) neighbourhood, a twee monstrosity of modern urban architecture. Blech.
During the devision of Jerusalem, 1948-1967, the Jordanians had the Eastern side including all of the Old City, and the valley marked a large stretch of the Jordanian-Israeli border. Snipers were posted on the Jordanian side to shoot down people in the aforementioned old neighbourhoods. It was reclaimed after the Six Day War and now a lrageinsh area of it is reserved for fairs, exhibitions, festivals and (mostly) live concerts. It's also a venue of the Jerusalem Film Festival, and movies are shown there, under the stars. Quite cool.