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.
Pros and cons of space telescope in an Earth-Moon L2-associated libration orbit
Some orbits will allow the possibility of spending some time behind the Moon which can completely eclipse Earth. That might be quite helpful for certain space-based radio telescopes. For frequencies above something like 20 to 50 MHz Earth's ionosphere is fairly transparent and terrestrial RF noise may be a problem. For frequencies below that Earth's surface is blocked but Earth's ionosphere is going to be noisy naturally.
For further reading about strange-looking libration orbits in the Earth-Moon system (especially those that spend time behind the Moon) refer to:
By the way, according to Wikipedia's Queqiao relay satellite:
Queqiao relay satellite... is a communications relay and radio astronomy satellite for the Chang'e 4 lunar farside mission. As part of the Chinese Lunar Exploration Program, the China National Space Administration (CNSA) launched the Queqiao relay satellite on 20 May 2018 to a halo orbit around the Earth–Moon L2 Lagrangian point. Queqiao is the first ever communication relay and radio astronomy satellite at this location.
I don't see eclipses of the Sun as a particularly bad thing. Eclipses are a big problem for SAR satellites observing Earth in LEO because they are very power-hungry (near-continuous multi-kilowatt radar beams) but a radio telescope receiver doesn't eat much. I think the biggest payload-specific power drain might be in the digital signal processing and digital recording which could certainly run on dedicated batteries for that purpose.
The Earth-eclipsing parts of the orbit would prevent communications or data links with Earth (ground stations or those in LEO) but not all the relay satellites in GEO, so I don't think there's a big downlink penalty.