There's a major problem with the way your coils are wound relative to the position of the magnets. I made some images to illustrate. These images show what happens with the voltage induced in your coil as they pass the face of a magnet. Coils are simplified to a single loop of wire for simplicity.
In this image you can see that the face of the north magnet is passing one side of the coil. A voltage is induced on half of the coil, and none on the other half, and you get an induced voltage, but not as much as you would have if both sides of the coil were working together.
The magnet continues to move and we have our second image:
Here you can see that both halves of the coil are about equal distance from the center of the pole face. As a result, the strength of the voltage induced in each half of the coil is equal. In addition to this, you can see that the voltages induced are pushing against eachother!(Both are pushing upwards toward the top of the loop). We can see in the center that the sum of these two vectors working against eachother is zero. The magnet continues to move past the coil and we have our third image:
Similar to the first image, the magnet is now passing by only one half of the coil and the induced voltage again has a sum effect, except now in the opposite direction. Note that as a result of the interactions you can see, a sine wave of sorts is still produced, even though we have only looked at what happens when the coil passes one magnet face.
Here's one last image to illustrate what you actually want to be happening:
This image shows a north and south pole working together on a coil, and you can see that because the north pole pushes up in the diagram and the south pole pushes downward, in the same direction along the loop, their vectors sum together for a much greater induced voltage.
So at this point you should see why your generator was producing close to no voltage, and you can see an awkward problem with the way your coil is wound. Because the coils are so narrow, both halves of the coil fit on one magnet face, so even if you had a complete ring of N and S magnet poles around your generator, you wouldn't get much induced voltage for the reasons I've explained above. (you would get more, volage, but not an even sine wave. You are using a single north and a single south pole opposite to each other, so you need to rewind your coils so that each half of each coil are on opposite sides of the spinning cylinder. As a result your method of using screws driven into a smaller center cylinder to support the coils is unlikely to work. You may wish to salvage a motor core off an old cordless drill(or other power tool) motor and do your windings on it. Brushed DC motors come with commutators fixed to the shaft that you can repurpose. Broken cordless power tools are cheap and have nice cores in them for hobby motor winding. Nice thing about using a factory made core is they are made out of laminate silicon steel to reduce eddy currents, and this will give you superior performance to any core you could build yourself. You can find small cores with a variety of numbers of slots, so you can wind 3 phase or single phase generators depending on the number of slots in your core. You probably want a core with an even number of slots so you can make a single phase generator to get started. I've rebuilt one 8 slot motor core into a generator personally. I wound my own coils onto the core and replaced the original magnets with rare earth magnets. It works really good for easy to build hobby stuff.
Sorry this answer is more than a year late, but I believe I've illustrated a fundamental problem that could come up in other hobby motor or generator builds.