Moving an object in an area
First you need to think about your axes configuration, then about the actual implementation. I for example can directly think of 2 options:


You could arrange your axes perpendicular to each other, like a Cartesian coordinate system. Here you might need more parts to build it, than with other arrangements, though it is a sturdy system and the positions and paths are easily to calculate. For implementation there are 2 common ways, both using stepper motors in most cases:


You could let the magnet travel on the 2 perpendicular axes pulled by a belt and stepper motor for each axis. As you have to mount one axes onto the other, you might need to use 2 stepper motors (one for each side) for the first axis (depending on your dimensions, how smooth the axes glide and how much force is introduced by dragging the figurine). This approach is mostly used by 3D Printers, since they need rather fast movement, which is also rather precise, but don't need to accommodate high forces.
Instead of belts and pulleys you could use threaded rods for each axis. The magnet is mounted on nuts on these rods. Rotating the rots will move the magnet over the axis, since the magnet is fixed in it's orientation. The precision is highly dependent on the used rods, but all in all it is often more precise, than belts. But it is also way slower.

You could use only rotational axes, like a SCARA robot. It has a rod on a rotational axis and another axis at the end of it. This approach might use less parts, than the Cartesian approach, but you need good servos to have precise and jitter-free positioning (though you could also go for a geared stepper motor here). Also the arm of the robot can look out under the field quite a bit, when positioning it to the far one side.




All of the above approaches can be way more precise, than you need it. They are all viable options.

I personally would go the first way, Cartesian axes with belts and pulleys. The principle is straight forward and you can easily hide it under your piece of plywood (including a margin for all the mechanical stuff of course).