Formula for how much power a motor generates
What would generate more power more wrapping of the coils or thicker
  wire? Is it based on total mass?


In general the more 'copper' (wire) and/or 'iron' (magnetic material) a generator has the more power it can produce. But deriving an exact formula is much more difficult than just weighing it.

The more efficient the generator is the more power it can produce without overheating, plus being more efficient makes it easier to crank. Lower efficiency means more heat, which may require a heatsink or fan which takes up room that could have been used by other parts.      

Thicker wire reduces the resistance that causes loss due to current flowing through the coils, but makes the windings larger which increases the size of the generator. The more turns you have the more voltage it can produce without cranking too fast, but this also increases size unless you use thinner wire, which has higher resistance. 

You can reduce the number of turns required by winding the wire around an iron core that concentrates the magnetic field. But this takes up room that could be used for copper, and introduces hysteresis and eddy current losses that increase at higher speed. Thin laminations of exotic grain-oriented silicon steel reduce core loss, but are expensive and difficult to manufacture. 

You could also use larger, stronger magnets, and try to get them closer to the windings. But there's a limit to how close you can get them without touching, and core losses also increase. Large high strength magnets aren't cheap either.     

So you must find a balance that produces the best result at the speed and size you want. The calculations required for accurate estimation of stator core size and shape, copper fill, magnet placement etc. are not easy because they involve 3-dimensional analysis of magnetic fields interacting with the various components. This is commonly done with a program called FEMM (Finite Element Method Magnetics). 

If you can't handle that then just try to get the thickest wire and strongest magnets you can you can into it, with the number of turns needed to get the voltage you want.