Every orbit in space has its pros and cons. Low Earth Orbit has accessibility but frequent eclipses whereas a Solar Lagrange Point is clear and stable but distant. In the case of the Lunar Lagrange Points (for argument we'll specify L4, L5, and L2 in the Earth-Moon system) what pros and cons exist from a telescope's POV?
LEO doesn't necessarily have to have frequent eclipses at all. In fact Sun Synchronous orbit never leaves the sunlight.
Contrasting to L2, L4, and L5 which all have eclipses. There's also the issue of stationkeeping. Lagrangian points are less stable then they appear; and keeping a spacecraft in one does require a good bit of fuel for stationkeeping. Lastly, there's an issue of data transmission. The spacecraft needs to be capable of returning high resolution photos back to the earth faster than it takes new ones; which requires increasingly large antennas and more power the farther from the earth you get.
By far the most fascinating orbit ever used for a space telescope has to be the one used for the Transiting Exoplanet Survey Satellite. It uses a highly elliptical 2:1 moon-synchronous orbit that goes pretty far out into space to make its observations, zips back close to the earth where it can achieve higher data rates, then back off to deep space it goes.
The Earth-Moon Lagrange points all have eclipses. Less frequently than Earth, but often enough to be a problem.
In addition, L2 is behind the Moon. This shields the telescope from Earth radio transmissions which would be a benefit for a radio telescope, but you'd need a communications relay satellite in a polar orbit around the Moon.