On planets, a space elevator goes to (geo)stationary[0] so that the cable doesn't wind up around the planet, but you can't do that on the Moon, because luna-stationary is occupied by the Earth, which inconveniently is too massive and spinning too fast to anchor the other side of the cable. However, the L1 point is also stationery relative to the lunar surface, and is the place where the gravity of the Earth and the Moon balance out.
[0] IIRC, geostationary specifically means Earth, but there's going to be some more general term for the same idea over generic parent objects and not just Earth
Lunar stationary orbit is around 88,400 km, which would be unstable for a satellite due to the Earth's gravity, but might allow for a space elevator pointed right at Earth to efficiently launch crates of helium-3 or hydroponic grain back to the planet.
Indeed. I've not even played with this in one of the many simulators, but I believe the suggestion is to put the counterweight a tiny bit closer to Earth than the L1 to stabilise it.
Although (and I wish I could find this again), I've read that lunar He3 is so diffuse that getting it out would incidentally give us so much purified aluminium, silicon, and oxygen, that sending all that back to Earth and magnetically decelerating it on arrival would give us more energy than the He3, as would burning those ingots with that oxygen.
Lunar stationary orbit is not 88,400 km, it's 384,000 km, i.e. the distance from the Moon to the Earth. The moon is tidally locked to the Earth.
Though you could equivalently try and go for Earth-Moon L1 with a counterweight on the other side of L1. It would be significantly more unstable though.
On planets, a space elevator goes to (geo)stationary[0] so that the cable doesn't wind up around the planet, but you can't do that on the Moon, because luna-stationary is occupied by the Earth, which inconveniently is too massive and spinning too fast to anchor the other side of the cable. However, the L1 point is also stationery relative to the lunar surface, and is the place where the gravity of the Earth and the Moon balance out.
[0] IIRC, geostationary specifically means Earth, but there's going to be some more general term for the same idea over generic parent objects and not just Earth