The Leo Stick is a clear acrylic tube with several coils mounted along its length. When a magnet drops through the tube you can see as a little pulse of light from the LEDs mounted on each coil. This happens because when the magnet passes through the coil (at high speed) the sudden change in magnetic flux through the coils causes a current to flow through the coils and the LEDs.
You can buy coils (if you can find the correct inner diameter) or you can wind the coils yourself. I wound my own coils using 22 guage magnet wire, and made about 10 layers of 35 turns for a total of about 350 turns of wire in each coil. Hold your coils together by brushing on epoxy glue. This is the way they do it at particle accelerator labs, so it is plenty good enough for us here.
Make sure you can recover the wires at both ends of your coil. You need to carefully strip the insulation off the wire and solder two LEDs in parallel between the ends of the coil. LEDs are light emitting diodes, so current will only pass one way through them. Therefore you need two LED's facing in opposite directions. When the magnet falls one way through the coil LED #1 will light up. But when you turn the Leo Stick over and the magnet falls the other way through the coil then LED # 2 will light up. I used LEDs that were recommended for 1.2 V. Test your coil and LED combination before you put everything together.
Put rubber stoppers in the ends of the Acrylic Tube to stop the magnet from falling out. I have not found a good way to hold the rubber stoppers in permemantly. You can be fancy and put springs at each end, but only if you can find springs that are Non-magnetic.