PCB fabricators will run electrical tests on boards by subjecting them to a "bed of nails" test (actually a much more complicated robotic approach these days).
The idea is that every net is tested electrically for continuity between different end-points. (Note that "continuity" in such cases is defined as "resistance below some threshold.") A comprehensive test might also inject test signals and measure impedance, capacitance, etc. to validate board performance.
A simplified version of this might be to use a digital multimeter with some probes attached to some sort of holder so that you can ensure consistent results between boards. (Meaning, you are removing variables like pressure and probe position.) If your budget allows for it, you might even consider something like the PCBite probe kit which is essentially a probing kit for small pins or just holding something like an oscilloscope probe to a given pad hands-free. (I won't link it because I am not affiliated and don't want this to come across as an endorsement.) You could probably make something to hold the board and probes in a repeatable way.
You would place a student board into a testing fixture, then connect the DMM probes to the relevant pads, and make a resistance measurement. You could also measure impedance if you have a function generator and oscilloscope, allowing you to send an AC signal along the track and measure it at another point. In all cases, you will need reasonable quality test equipment — an inexpensive DMM may give you acceptable results, but an extremely cheap one will be inaccurate and just give you headaches.
If your students are making variable-width tracks or having differing results etching copper, this may be a usable objective way to quantify track performance. I'm not sure what processes you are wanting to check; if your students are trying different substrates, then you might need more advanced tests to measure how that affects performance. (A copper track on fiberglass versus, say, plastic, will perform the same at DC, all else being equal.)