Najwa - I think you understand the experiment a little better now. The original author of the paper, from which this project was created, explains in excellent detail what is happening in the experiment. As he states, the generator will only produce power as the coil passes through the flux field of the magnet. Therefore it will only illuminate for a brief period as the magnet moves past the coil. In addition, he describes how to build a storage device that will enable constant illumination of the LED. Here are the links to these pages:http://www.creative-science.org.uk/gensimple1.html
Toward the bottom of his paper there is a link to building a storage device. Click on that for details. He has described a rectifier circuit with a high value (10,000 microfarad) electrolytic capacitor to provide a power storage unit. He correctly recommends the use of germanium diodes for the rectifier, but I think you may have problems finding them at a Radio Shack. Rather than a 4 diode 'full wave bridge' I would suggest a simple half wave rectifier using a single Schottky diode that has a low forward voltage drop. These seem to be available at Radio Shack as their part number NTE584 for only $0.68. If you can find a germanium diode (1N270 series, for example, it's probably even better). At Radio Shack you can try to find the lowest cost, highest value electrolytic capacitor, with under 10 volt rating. I found their part number NTE VHT3300M10 a 3300uF, 10V part, for $1.52. It should work fine.
I've attached a sketch on how to wire it up. Because we are now dealing with DC, you will have to be mindful of which way the LED is connected, also the polarities of the diode and electrolytic capacitor must be connected properly. It is probably difficult to tell the polarity of the LED so just try it in one direction to see if it lights. If it doesn't, reverse the leads. There is no danger of harming the LED with these low voltages.