For me, knowing Maxwell's Right-Hand Rule doesn't help much. It may seem superficially simple because there are only two possible answers. It's hard because there are so many binary elements that can potentially be flipped:
Wire coiled clockwise or anti-clockwise?
Clockwise when looking from above, or clockwise when looking from below?
Current flowing top to bottom or bottom to top?
Are we talking conventional current flow (ie, positive to negative) or actual electron flow (the opposite)?
Is the north end the end that seeks north, or the end which has the same polarity as the north pole?
Left or right-handed screw (or hand, in some accounts)? (Maxwell has both kinds of rule, one for generators and one for motors.)
Whether the needle picks up the same polarity as the coil, or the opposite polarity?
Knowing a bit more about the subject arguably makes it harder to remember. The difficulty is forgetting the possible accounts/diagrams which are wrong. I think I now know the answer, but I'll probably have forgotten it within a week, because it is all just too arbitrary to keep in my mind. (And further elucidation probably won't help and isn't requested.)
Its only schoolboy physics, so it is not really that difficult to get right.
As long as you remember a few conventions.
Rule 1) - Current flows from the positive to the negative battery terminal.
Rule 2) - Magnetic flux lines flow from North to South.
This diagram is useful to remember as is shows a sectional diagram of a magnetic coil. Each circle with a dot in the centre shows current coming towards you out of the diagram and each circle with a cross shows current going away into the diagram. This is just a simplified interpretation of Maxwell's right hand rule.
The magnetic flux lines now indicate the magnetic south polarity is on the left hand side and the magnetic north polarity is on the right hand side i.e. as above in Rule 2.
Now that the north south polarity of the coil has been established from first principles, all that is required now is to determine which direction the sharp pointy end of the needle ends up. Will it be at the North or the South end of the coil.
This is pretty simple; for the sharp pointy end of the needle to point north in the earths magnetic field then the sharp pointy end of the needle has to be magnetised as a south pole. So therefore the sharp pointy end needs to pointing to the south pole of the coil i.e. the left hand side of the above diagram.
Of course if folks cannot tell their right hand from the left hand (you'll be surprised how many youngsters can't) then knowing the above information won't make much difference.