Actually I'm not sure if a base on the Moon or "hovering" in the L1 or L2 positions is better - depends on how human body reacts to lunar gravity for a long term stay - which is unkown.

If we need artificial gravity - that might be easier to achieve in orbit with a tether system. Though if we just need, say an hour or so of gravity a day and can do that in a short arm centrifuge then surface could be fine.

If we do need full g or close to it 24/7 then the best solution for the Moon involves things such as a habitat permanently driving around a circular track on the moon to generate artificial gravity - or a spinning habitat - depending on what radius and spin rate humans can tolerate. Those may be practical - but it might be easier to use a tether system in space if that's what we need to do.

And in both cases I think we will spend most of our time inside the habitats shielded from cosmic radiation. Because - who wants to add even a 1% or 10% chance of getting cancer in your lifetime - when the cancer would reduce your lifespan typically by a decade or so? And spacesuits and lunar rovers can't be shielded to prevent that risk at least with present day technology. And spacesuits are clumsy, and though both telerobots and spacesuits increase in capability I think telerobots will have the edge on the Moon in the near future, maybe even already. For instance if Apollo 17 used present day telerobots - they wouldn't have been limited for safety reasons to within a walking distance of the lunar base - and could have spent far more than the 7 hours spread over 3 days on exploring the Moon. And the Apollo astronauts had a fair bit of trouble doing simple tasks such as picking things up from the surface - telerobots might well do these better than humans in spacesuits.

Anyway - doesn't make much difference, both ways are many exciting possibilities.

• Build an infrared telescope in the "craters of eternal night" passively cooled - it's the coldest place in the inner solar system
• Search for ancient meteorites from Early Earth on the Moon. These may be buried deep below the surface, some of them - and be unchanged since the first few hundred million years of our planet. And later on - must be meteorite fragments on the Moon from later impacts - including the one that lead to extinction of dinosaurs. And - they have surprisingly soft landings in the low Moon gravity - and material at the edge of a spreading impact crater on the Earth is only lightly shocked - is not impossible that we get e.g. fragments of ammonites and dinosaur remains etc on the Moon. And good chance of eventually finding early organics and remains of early cells even the Last universal ancestor  and earlier Ur-organisms
• Study the ice layers at the lunar poles - like Antarctic ice - they may have a record of the solar system - of volatiles and other hard to preserve things. Including again quite possibly early buried meteorite fragments and dust with organics.
• Explore the Moon - it is not at all well understood especially geologically. Only had one visit by a geologist so far - on the Apollo 17 expedition. Only a limited amount you can learn from orbital surveys.
• Build liquid mirror telescopes within lunar craters
• Build radio telescopes on the lunar far side which would be able to study the sky over the entire radio spectrum
• Learn how to do telerobotic exploration of the solar system in a place close enough to Earth for return to Earth or resupply in case of disaster
• First attempts at closed habitats and space settlement (either on the Moon or in lunar orbit) - and prove that humans can survive for years on end in space without resupply from Earth. If this can be done, then entire solar system is open to human exploration eventually. If not - then we have to remain in the Earth Moon system and explore the solar system with autonomous robots instead - at least until it is practical to travel through the solar system in weeks and days rather than years.

Of those, personally I think the search for ancient Earth life on the Moon is the most exciting in its potential implications. With - far side radio telescopes - another very exciting possibility.

For more on this, see my new Case For Moon - Open Ended Positive Future For Humans Based On Planetary Protection - Executive Summary

and in much more detail:

Case For Moon - New Positive Future For Humans In Space - Open Ended With Planetary Protection At Its Heart