Let's share.
In the meantime, here are some of the suggestions by NASA.
Radio astronomy
Put a radio telescope on the far side of the Moon so that it would
be shielded from Earth's copious radio noise, and would be able to
observe low radio frequencies blocked by Earth's atmosphere.
Observe at these frequencies and there may be a chance of it
leading to many exciting new discoveries.
Study the cosmos in greater depth
The moon would also be an excellent place to study the high-energy
particles of the solar wind, as well as cosmic rays from deep
space. Earth's magnetic field and atmosphere deflect many of these
particles, so even satellites in low-Earth orbit can't observe them
all. The moon has virtually no atmosphere, and it spends most of
its 28-day orbit outside of Earth's magnetosphere. Detectors placed
on the moon could get a complete profile of solar particles, which
reveal processes going on inside the sun, as well as galactic
cosmic radiation from distant black holes and supernovas.
Bonus: These particles are trapped by lunar regolith, the layer of
crushed rock and dust covering the moon's surface. This means that
lunar regolith contains a historical record of solar output: core
samples could tell us about changes in solar output over billions
of years. "We believe that the moon's preservation of this solar
record is unique and can provide us with insights on how past
fluctuations in the solar output have affected, for example, the
history of life on Earth," says Volosin. In particular, it could
shed light on the extent to which solar variability and galactic
cosmic radiation influence climate change.
Some other suggestions
Lunar power from solar cells, protected data archives, mining of
lunar metals, and research under conditions of low gravity and high
vacuum.
Full list of things to do.
