The “prime meridian” is the line through the black holes at the centers of the Milky Way and Andromeda, and the “equator” is the galactic plane.

Up. North on a map is always up.

Within the Milky Way a polar (cylindrical) coordinate system makes more sense than Cartesian - there's an axis of rotation to define the center and 'up/down' directions. Zero degrees is arbitrary but a line from the galactic center to Sol, projected onto the galactic plane, would be an obvious choice as a sort of galactic prime meridian. 'North' and 'south' don't really map to a roughly disc shaped galaxy - you'd use distance from center, angle, and 'elevation'.

On an intergalactic scale, the center of our own galaxy is probably still the obvious choice for a center point. We could use the same axis and meridian - I don't think the rotation of our galaxy matters on any human timescale, and on the time scales where it does matter, everything is moving relative to each other so coordinates already aren't 'fixed'. I'd use a spherical coordinate system instead of cylindrical for intergalactic coordinates, since things are not roughly in a plane anymore.

If you want a fixed coordinate you'd have to include a time dimension, and as the zero point for time I propose the Unix epoch. Not because it makes any sense but because it's extremely funny to imagine computer systems in the year 10000 still relying on that legacy decision. Though special relativity makes 'point in time' rather complex as well - I don't know enough to know what you'd actually need to make that work.

Of course we already have such coordinate systems for astronomy if you want to know the 'real' answer, one of them is pretty close to what I just came up with: https://en.wikipedia.org/wiki/Astronomical_coordinate_systems

SagA*

It's as good a place as any other.

Larry Niven kind of works out this naming in several of his novels. I don't remember all the specifics, and he also used a similar scheme to describe travel in ring world, but it's close enough. First, don't bother with calling it north, that is just confusing. In the reference frame of yourself or the map you're drawing in a spinning galaxy, you've got spinward (in relation to the galactic spin) and anti-spinward, in (toward galactic center) and out, and then normal (orthogonal) to those dimensions, which you could call up and down depending on your preference. I'd probably call spinward, inward, and up positive.

If you need a reference (north) for a galactic map, it's really unlikely you'll want to use anything like grid coordinates. You can use the same ideas from the local map. You'd probably want an origin at the gravity center of the galaxy and pick another object as a reference point from which to zero angular measurements around the disc. That other object could be another galaxy (if you want to measure galactic spin itself) or something distinct and obvious in our own galaxy (if you want to navigate within the galaxy). Most civilizations would probably just use a line between their home system and galactic center as their prime meridian. Up and down should be orthogonal to spin again. If you're home planet had a magnetic pole roughly pointing out of the galactic disc (like ours), you'd probably choose your "north" pole's side up.

Well, let’s just try where a compass points to in space?

The rotation axis of the Milky Way. It seems more reliable for me than to pick two astronomical objects, as their direction might change over time. Plus it makes intuitive sense considering how we did it for Earth.

Pointing at the north pole of Earth. You didn't stipulate any other spacefaring species, so it's not like there'd be competing standards.

It would point toward the galactic plane/center of the system.

Tons of space video games do this and it actually helps orient yourself really well.

I think you would pick a north depending on your place of origin. Probably from either their own planet or, more likely, their own galaxy.

But when you get to another galaxy you'd probably have to pick again. Galaxies aren't all oriented the same way but they do roughly spin at the same speed. So, just like earth, from a certain point of view everything spins around the north south axle. But that does not have to line up to your home galaxy. And even if they did, it might spin the other way around. Much like venus spins in the opposite direction of the rest of the planets in our solar system.

What I'm trying to say is that using a universal North is completely impractical for intergalactic space travel. So you'd probably use a local system. And where that's not possible, you'd probably use your point of origin for orientation.

North is the right hand rule to galactic spin.

The location of our solar system. Everything would be in relation to where our solar system is in relation to where you are.

North, duh

M87

I'd use Earth.

I would just spin a quantum gyro, and let it decide...

We could keep using Polaris or find some really far galaxy or star as the north.

Easy, the sun is the bright spot and anything else is too far away to even begin to think about exploring, so the Sun is a good point of reference for early explorers. Once you journey further than our solar system, you'd need another point of reference and it would most likely be the solar system as a whole.

The Hercules Corona-Borealis Great Wall.