How much does the power of a magnet increase voltage on electromagnetic induction?
Yes, double the Gauss will double the output voltage, but only if the shape of the magnet remains the same.  [SAFETY WARNING: for little kids, neodymium magnets wider than 15mm are a finger-pinch hazard.  As a result, many DIY hobby projects avoid using neodymium magnets.]

Anyway, it's much easier to instead double the number of turns of wire, use 600 turns, or even 1000.

One possibility is to make two changes, not just one.  Change the magnets and generator shape, and also use Neodymium rare-earth supermagnet type.   I made a simple generator which only needed 40 turns of wire, not 400, and it would light a red LED when spun by fingers.

Get a toy "buzzing magnets" or "rattlesnake magnets," the larger jumbo version 60mm long.  These magnets are magnetized crossways, with the poles on the sides.  A generator would spin them on axis, NOT flipping end-over-end.  So, the generator can be inside a plastic pipe, and the wire coil is VERY close to the magnet.  We only need one of the two supplied "ellipsoid egg" magnets of course.  (Or, make two copies of the generator!)

To easily spin the magnet, give it an axle using two nails.  I used epoxy to glue one large nail head-first against each end of the magnet.  Carefully align the nails while the glue is setting, so they are on axis, nails pointing away from the ends, so the magnet will spin very smoothly.  Or you can cheat: first glue each nail with a dot of Crazy Glue.  That way, if the magnet wobbles too much when spun, you can break them off and try again.  Then, if they're well aligned, add a bigger glob of 5-min epoxy glue to make it permanent. 

Then I located a plastic tube at a hardware store, just wide enough so the magnet would fit inside very close but without touching.  (I think this was 5/8in. grey electrical conduit pipe.  Or maybe 3/4in?)  Cut off a segment of the pipe which is slightly longer than the magnet-plus-glue-globs-nailheads.

For the axle support end-plates, I cut out two rectangles of acrylic sheet, 1" x 1.5", and drilled a hole in the center of each.  Make the hole just slightly larger than the nail diameter.  I put the magnet in the pipe, then aligned and glued both plastic plates, but only with tiny amount of epoxy, so I could tear this open if needed.

Verify that you can spin the nail and the magnet very fast and smooth inside the pipe.  Perhaps put a tiny bit of grease or oil into the hole in each plastic plate.  Drop some paperclips or tiny screws on the pipe and see the crazy motion!

Now wind some enamel "magnet wire" around the end-plates, winding lengthwise along the tube.  DON'T wind it around the pipe circumference, that won't work.  Wind it lengthwise along the pipe, crossing over the end-plates and almost touching the nails.  Wind half on one side of each nail, half on the other, so the coils aren't rubbing against the nails.

First try only 100 turns of wire, see if that's enough to light a red LED.  If so, start removing wire!  See how few turns are still able to light the LED when spinning the nail with fingers.  Or instead, add more than 100 turns of wire, and see if your generator can produce enough voltage to light a green LED, or even blue or white.

WEIRD: in theory, if you build two of these, then operate them very close together, the voltage of each will be much higher.  The spinning magnets lock together like "invisible gear teeth," and they produce a very large field in the space between them.  This is much the same as putting one of these plastic-pipe generators inside a close-fitting iron pipe.  In both cases, any flux which had been running across the magnet surface and missing the wires, that flux will be "pulled out" by the nearby magnet (or nearby iron surface.)  A similar voltage-increase would occur if you could slide the wire coil across the magnet surface, rather than having a few-mm gap of plastic pipe-thickness and air space.


A bit of info is on coilgen project page.
Sources of magnet-wire are listed too.
Here's another one: neo magnet shaker generator