Increasing magnet strength, disc diameter and number of magnets does effectively increase power.
Cut in voltage depends on the rate of change of flux and the number of turns. You can get any cut in you want by altering the number of turns if all else stays the same.
Power eventually depends on the voltage and circuit resistance. you can only get more power by lowering the circuit resistance once you fix the cut in.
The way to lower resistance is to use thick wire, which restricts the number of turns. Stronger magnets, more of them and increased frequency are the things that let you get more volts with less turns.
Larger discs let you get more nagnet area in, you can do it with few magnets of large area or lots of smaller magnets but by adopting lots of smaller magnets you keep the turn length short and it is easier to get the same result.
Flux
