While the OP never clarified the question themselves, edits and comments suggest that what is wanted is an orbit that leaves an artificial satellite roughly above a specific site on the Moon's surface and visible from that point, analogous to how a geosynchronous satellite remains roughly above a specific site on the Earth's surface.
@Russell Borogove's answer which invokes @zephyr's answer in Astronomy SE is correct if one a priori requires the central body being orbited to be the Moon.
However:
tl;dr: There are two potential solutions to this problem but neither is an orbit around the Moon per se.:
- Exploit three-body orbits
- Attempt to use a Moon-synchronous Earth orbit.
1. Three-body orbit around an Earth-Moon Lagrange point.
EM L1: A Halo orbit associated with Earth-Moon L1 would keep a spacecraft between the Earth and the Sun and fairly close to the Moon. L1 (and L2 below) fall roughly at the Hill sphere radius $R(M_{Moon}/3 M_{Earth})^{1/3}$. With a separation $R$ of 385,000 km and a mass ratio of about 1/81 that would be about 60,000 km above the Moon's surface and meander around that area by several thousand km.
Long term it wouldn't be stable and requires some station keeping, but of course geosynchronous orbits around the Earth also require station-keeping to keep from drifting east-west to be over a substantially different point on the Earth, as well as to maintain low inclination.
EM L2: Similar to EM L1 above except visible to the far hemisphere of the Moon.
This orbit is currently being used by the Queqiao communications satellite, see below.
EM L4 & L5: similar. Since the Earth/Moon mass ratio of about 81 is greater than the critical value of $(25 + \sqrt{621})/2 \approx 24.96$ one can put spacecraft in orbits associated with these points, which will remain roughly above points 60 degrees east and west of the average sub-Earth point on the Moon's surface.
All station-keeping caveats apply here as well.
The title of the question was originally What is the altitude of a Moon synchronous orbit? so I'd started down this path before clarifications were made for the OP by third parties.
Results: nice try but doesn't work.
While a high inclination Sun-synchronous orbit around the Earth can always keep the Sun in view by slowly precessing around the Earth (once per year), there is no solution that could precess once per lunar month of about 27.3 days.
The ISS precesses around with a period of the order of sixty days for example, but at it's inclination it is usually (but maybe not always) hidden from the Moon once per orbit.
Queqiao remained visible continuously both to Chang'e 4 and Yutu 2 on the far side of the Moon and to Earth

Source: Wikimedia Commons by Loren Roberts for The Planetary Society https://www.planetary.org/space-images/change-4-mission-profile