From the article you referenced:
The peak voltage generated by this little device is given by:
V = A x M x N
Where A is the cross sectional area of the can (0.0008 m²), M is the rate of change of magnetic field (ca. we need to use very strong magnets having a surface field of say 1 Tesla (see magnet info. below), so shaking it say 5 times a sec we get M = 5 Tesla / sec) and N the number of turns.
If we want an LED to light brightly we need to generate peak voltages of about 4V;
rearanging the formula allows us to estimate the number of turns:
N = V / (A x M) = 4 / (0.0008 x 5) = about 1000 turns - happy winding !
If I'm reading this correctly, the size of the magnet is not important, it's the cross section of the container holding it. If you want to reduce the number of windings, I suggest increasing the rpm of the magnet.