How to move a magnet to a specific coordinate on a board
Possibly the easiest way to do this with off-the-shelf or easily-made parts is to have a single permanent magnet moved in two dimensions using linear positioning mechanisms, which looks like what the original mouse did. 3D printers do exactly this in one form or another — you could just use parts from a 3D printer's X-Y or X-Z axis sets.

Applying a variable magnetic field field from individual coils around the edge is possible, but it is a very hard control problem because the "mouse" will have unpredictable friction and the force exerted depends strongly on the distance to the coil — you will need to have very good low-latency tracking of its position and a good bit of control theory knowledge, and very strong electromagnets (with serious power electronics for drivers) to work over that distance.

You could use a Helmholtz coil to simplify the problem by making the magnetic field approximately uniform within the volume of the maze, but they would still need to be very strong.

If you use multiple coils under the board, you can avoid having to wire up and control many individual coils by using a repeating pattern of "phases", like a stepper motor unrolled flat (e.g. see https://hackaday.com/2018/01/08/a-stepper-motor-for-two-dimensions/ ).