Btukfyl

Assuming we had a non-spherical asteroid that doesn't have a magnetic "north", how would the inhabitants define areas on the asteroid? How would they explain to a visitor to go to a very specific spot to retrieve or leave something besides "head over the hill sunward for 50 km"

My suggestion would be that you select a point on the asteroid to act as a pole. Perhaps the point of first landing? Then, using that point and asteroid's centre of gravity as references, you can map spherical coordinates.

Whoever is going to be on that asteroid will necessarily used radio communication to keep in contact with the rest of the crew.

To ensure communication a network of antennas has to be established, since a single antenna could at best serve half of the asteroid.

Each position can then be simply referred to the distance from the (closest) antennas.

I'd maybe consider using the axis of rotation - it would be a very rare asteroid that isn't rotating somehow. Imagine sticking a skewer through the asteroid along the axis. That would give you a top and bottom, and then you can use spinwise and counter-spinwise (or something similar).

Obviously only works if you have an asteroid that is rotating nicely, something that is rotating a bit more chaotically might be more of an issue. If it's not rotating at all, or is spinning chaotically then Arkenstein XII's answer is definitely the way to go.

You can use the same coordinate system for celestial navigation (latitude and longitude) used by Terran mariners. What you need is : an almanac, a watch, a device for measuring the angle of the stars relative to some average horizon, and a map.

Almanac

The basic concept of celestial navigation is this : imagine several easily-recognizable stars. Next, imagine that you could draw a line from each of these stars that would pass through the center of whatever you are standing on. This line will touch the ground at one (only one) location. An almanac records these stars and the location (in latitude and longitude) of the point on the surface where the imaginary line from the star touches the ground.

Watch and Calendar

And, this spot will move as the object rotates around it's own axis (days); but will only move a little with the seasons.

Measurement Device (Sextant)

When you are standing on the spot where this imaginary line from your easily-recognizable star intersects the ground, that star will be directly overhead.

Map

Likely, you are not standing on one of these spots at any particular time. The angular measurement times the average radius of the asteroid provides you with the approximate circular (radial) distance between that point and where you are. Measure multiple stars to determine where these circles overlap on a map. That is your (approximate) location