Would a station in a stable Lunar Cycler orbit have any large benefit over a LEO or HEO orbiting station?
I know the mission-to-mission cost of getting to/from such a station would be more but I imagine the benefits to the kinds of things done on the station would be more interesting and possibly outweigh the costs.
Major benefits:
A lunar cycler allows you to use smaller spacecraft to get onto a larger spacecraft. The smaller spacecraft could be tiny, as long as it can rendezvous with the cycler. So a spacecraft that is cheap to launch, or refuel, but not suitable for long journeys to the moon, could be used to ferry people. This could have the big benefit that the launcher required to get to the moon could be smaller for a larger living space/lab space. However, as you probably know, you'd still need the delta-v to be able to get to the moon, as you'd essentially be matching orbits with a lunar trajectory anyway.
For example, the cycler might have a lot more radiation shielding than the transit craft, but since you are only in the transit craft for a few hours it's OK.
Minor benefits
A cycler would be interesting from a long-term space habitation study, as it would allow you to go back and forth between the radiation belts a bunch of times. Admittedly this could be done in an Earth-orbital station as well.
A cycler would make for an interesting tourist experience - go up in a small, cramped spacecraft, cruise to the moon and return in style, then land in the same small craft.
Ultimately, a lunar cycler may not be the most relevant, but a Mars cycler would be amazing. You could send humans in a tiny vessel just large enough to get them onto the cycler, then re-use the cycler for multiple missions (I believe this is what was done in The Martian). The idea is to get it set up once, then forget about it, rather than re-launch the transit vessel each time.
A cycler has one big drawback compared to a LEO station: it'll be outside the Van Allen belts much of the time, and if its closest approach to Earth is at low altitude, it'll go through the belts on every orbit. Both lead to a massive increase in radiation.
