How to measure the "efficiency" of a generator?
Measure the mechanical work input and the electrical work output.

It sounds like you might be able to drive your generators mechanically using an electric motor with a crank and connecting rod to move the magnet like a piston.  You could take comparative measurements by measuring the electrical power input (for example, current while running from a voltage regulated supply) and on the output measuring the voltage across a load resistor small enough in value to increase the current draw of the motor when connected, but physically large enough not to heat up (as its resistance might then change).  This would be a comparative measurement only with which you could compare your different generators, as your calculated efficiency would be the product of your driver efficiency (which you wouldn't know) and the efficiency of your test generator.  Arguably you need to run the driving motor at the same speed for all the tests, or have a plausible model for how its efficiency should vary with speed if your test generators have different speed requirements.

You could try doing something directly physical, for example a mass on a string winding or unwinding from a drum.  In the real world this doesn't work as well as it does in physics textbooks though - inertia can overwhelm the continuous effect you are trying to measure, and the mass tends to want to swing side to side and bump into your knees or workbench, instead of just ascending/descending cooperatively.

Finally, consider that ultimate efficiency may be less important than effective utilization of the source of mechanical power available.  For example, if you are building a shake flashlight, you probably want either the design that is the brightest, or the one that provides just enough light to see by for the least vigorous (or frequent, if you have caps in there) shaking.