Phobos may be the first place to actually have a Space Elevator.  Phobos is the larger and nearer of the two small moons of Mars, the other is Deimos.
http://en.wikipedia.org/wiki/Phobos_(moon)

The low gravity (0.0084 – 0.0019 m/s²) and low escape velocity (11.3 m/s = 40 km/h) of Phobos means that it could become a staging point on journeys between Mars and the Earth.  The interplanetary transfer vehicles would land on the moon and the astronauts transfer to Mars landers.  The spacecraft being parked in hangers by the underground Phobos Base.  Building the Base underground is a simple way of protecting its inhabitants from the high levels of radiation in space.  The moon may be mined to obtain water and propellant for the spacecraft.

Phobos is not a sphere (26.8 × 22.4 × 18.4 km) which causes some interesting effects including the Mars Phobos Lagrange points L1 and L2 are only 2.5 km above the surface.  The Space Elevator could be made from current materials like steel, aluminium or Kevlar.  By extending the ribbon so that it becomes a tether a spacecraft climbing it can go directly into low Mars orbit.  A 6153 km tether can supply sufficient delta-V to insert a spacecraft into the Hohmann transfer orbit to Earth.
Ref http://forum.nasaspaceflight.com/index.php?topic=18812.0