I think a large part of the justification for a moon base is that it could be used as a location from which to stage further space missions. If you construct a rocket on the moon you've got no atmosphere and a fraction of the gravity to deal with.
No atmosphere might also make it useful for astronomy, but I'm not so sure of that.
A lunar base would provide an excellent site for any kind of observatory. Particular advantages arise from building observatory facilities on the Moon from lunar materials. As the Moon's rotation is so slow, visible light observatories could perform observations for days at a time. It is possible to maintain near-constant observations on a specific target with a string of such observatories spanning the circumference of the Moon. The fact that the Moon is geologically inactive along with the lack of widespread human activity results in a remarkable lack of mechanical disturbance, making it far easier to set up interferometric telescopes on the lunar surface, even at relatively high frequencies such as visible light.
A lunar base could also hold a future site for launching rockets, to distant planets such as Mars. Launching rockets from the Moon would be an easier prospect than on Earth due to the Moon's lower gravity requiring a lower escape velocity.
No atmosphere might also make it useful for astronomy, but I'm not so sure of that.
EDIT: Wikipedia, of course, has a section on this: http://en.wikipedia.org/wiki/Moon_base#Advantages
These points on their list:
A lunar base would provide an excellent site for any kind of observatory. Particular advantages arise from building observatory facilities on the Moon from lunar materials. As the Moon's rotation is so slow, visible light observatories could perform observations for days at a time. It is possible to maintain near-constant observations on a specific target with a string of such observatories spanning the circumference of the Moon. The fact that the Moon is geologically inactive along with the lack of widespread human activity results in a remarkable lack of mechanical disturbance, making it far easier to set up interferometric telescopes on the lunar surface, even at relatively high frequencies such as visible light.
A lunar base could also hold a future site for launching rockets, to distant planets such as Mars. Launching rockets from the Moon would be an easier prospect than on Earth due to the Moon's lower gravity requiring a lower escape velocity.