Hi,
Just a couple tips...
1. It would probably work best to use a cylindrical magnet that could slide up and down in the tube, although it may be hard to find one that fits really good.
2. A thinner wall tube is better as there is tighter coupling between magnet and coil, although the magnet has to fit the tube good also.
3. The more turns of wire the better, but the gauge has to allow the required current flow without too much loss.
4. You probably have to use a bridge rectifier to rectify the AC output of the coil into DC for use as lighting an LED.
5. It would probably help to have a capacitor to store some of the energy generated.
6. Some bumper padding at the ends to reduce the collision force at the ends, but even better some medium strength springs which would store some of the mechanical energy and help get the magnet moving back in the other direction. This means much less effort on the part of the human for generating a given amount of energy and could make a big difference if the springs are chosen carefully.
Perhaps others here can come up with some more ideas too.