NavList:

To start with, I made an elementary experiment using this web site:

https://www.ngdc.noaa.gov/geomag/calculators/magcalc.shtml

which tells me the compass declination at a given location. At my approximate location 40N 87W the declination is 4.40 deg (W) with error 0.37, changing 0.04 per year. 

Now 1 degree West of my location, the declination is 3.67 deg. The difference 0.73 deg=1d20'  with half degree error, so I conclude that in principle, the method is usable to determine longitude at my location is barely within the error specified in the conditions of the Longitude Prize.

Now looking on the map, https://en.wikipedia.org/wiki/Magnetic_declination#/media/File:World_Magnetic_Field_Model_2025.jpg

we find the places  where the lines of equal declination have approximate N-S direction. This will be is optimal for the method. A brief inspection of the map shows that N America is exactly such a place.

So the method is not sound, in principle, in some locations. I do not address the practical problem of reading the compas to a fraction of degree  actual determination of variation which is usually done by astronomic methods.

Another thing is that declination changes with time, and one needs either to make periodic surveys of large areas of the Earth,or to have a mathematical model which predicts this change. None of this was available at the time of Halley.

Alex.