Much of this answer comes from the NASA publications Apollo Lunar Descent and Ascent Trajectories and Programmed Guidance Equations for Luminary 1C.
It turns out this is really two questions:
Could the Apollo Lunar Module have landed at an arbitrary location and returned without Earth assistance?
Without a doubt.
Descent was divided into four phases:
Descent Orbit Initialization. This was a brief retrograde burn performed 180 degrees opposite the point at which the landing burn would start. The purpose was to efficiently lower the LM's periapsis so that it wouldn't accumulate too much vertical velocity during the braking phase. For all practical purposes, this is a classic patched-conics maneuver which can be calculated using pencil and paper and the vis-viva equation -- or since the LM is in a nearly-circular orbit, the crew could just perform the already-calculated maneuver half an orbit early or late.
Braking phase, done using program P-63. This is the longest part of the landing burn, mostly reducing the spacecraft's horizontal velocity (some vertical-velocity reduction is done, particularly towards the end of the burn). Guidance here is completely autonomous: the guidance computer knows where it is and where it wants to go, and adjusts the throttle and spacecraft orientation to reach it.
Approach phase, done using program P-64. This is nearly identical to the braking phase, just with different targets. This is flown in a more upright attitude, so the crew can see where they're going and adjust the the flight path (or abort) if needed.
Landing phase, done using program P-65 (fully automatic), P-66 (semi-automatic), or P-67 (fully manual). In practice, every lunar landing was done manually, and an autonomous landing would be no different.
The targets for the braking, approach, and automatic landing phases were pre-computed on Earth. The key thing here is that these targets are expressed in the "descent coordinate system": a coordinate system centered on the desired landing point and oriented relative to local vertical. Compute a new landing point in Moon-centered coordinates, adjust a few time targets (such as braking-phase engine ignition) to match, and you can land wherever you want. The math is a bit complicated, but doable if you've got some way of computing trig functions.
The only real problem is if you guess too wrong about the elevation of your new landing point. The descent system can handle an error of plus or minus 13,000 feet at the point where the landing radar starts seeing the ground, but the Moon's got more elevation variation than that.
Ascent? Not a problem. Since the ascent system is used in the event of an aborted landing, it's capable of guiding the ascent module into a lunar orbit from a very wide range of situations, completely without Earth-based guidance (one of the abort scenarios envisioned was complete loss of Earth communication). Launching from the wrong part of the Moon is no problem: start the ascent program (P-12 if you're launching from the surface), and it'll put you in a 45-by-9 mile orbit roughly co-planar with the command module, with the periapsis eight degrees prograde from your landing site.
If you time the launch right, you'll be in a position where the rendezvous radar can see the command module. If so, the guidance computer can calculate and perform a rendezvous (programs P-32, P-33, P-34, P-35) followed by manual guidance and docking. If not, better hope the CSM comes into sight before you run out of electricity or cooling water.
The main problem with landing without Earth assistance is a loss of safety margin: to reduce the crew workload and the weight of the LM, a great deal of monitoring was offloaded to Earth-based facilities. For example, Manned Space Flight Network tracking provides a third trajectory estimate to permit two-of-three malfunction checking on the guidance computers. Additionally, landing at an un-surveyed location carries the risk of slamming into the side of an unexpected mountain, a hazard that struck at least one of the Soviet Union's landers.
Could the crew of Apollo 12 have done so?
Not without considerable preparation on Earth. The Apollo Guidance Computer had no routines for updating the descent coordinate system from the DSKY (the onboard user interface). The only method available for the Apollo 12 crew to update it would be by poking raw values into memory locations, something that requires careful study of the program source code.
