Step 1: What You Need:
2 x White or blue LEDs (25mA or better)
1 x Silicon diode (choose a schottky like the 1N5819 for better performance)
1 x BC327-25 or MPS751 PNP transistor (Do not substitute)
1 x BC337-25, MPS651, 2N4401 or PN2222/MPS2222 NPN Transistor (Do not substitute)
1 x 220uH coil inductor (see text)
1 x 10uF 10-volt (or better) capacitor.
1 x 1000pF (1nF, 102) capacitor.
1 x 10k resistor
1 x 1M resistor
9-volt battery and mounting hardware.
Most of these parts are available at AllElectronics Surplus and on eBay, here and here
If you want to experiment, you can try and make your own coil by starting with 15 turns on a toroid. If the circuit stays on by itself, add more turns; if the LEDs are dim, REMOVE turns.
Step 2: Assembly
The first picture below shows how I arranged the components. The next image shows the wiring on the underside. The third image is the 2 superimposed together.
Step 3: How It Works
The circuit begins operation when moisture from your finger turns transistor Q1 on, which switches Q2. Current from the battery is drawn through the 2 LEDs, the coil, through Q2 and to Ground.
C, the 1000pF capacitor forces Q1 and Q2 to saturate - and improve efficiency. Since the 2 LEDs only require 6.5v to light up, the coil builds up its charge with the excess of (9 - 6.5 =) 2.5-volts.
Capacitor C cuts off the transistors when the coil saturates and its magnetic field collapses. This supplies voltage which flows through the diode to keep the LEDs lit.
The oscilloscope trace shows how this cycle repeats over and over, almost 1/4-MILLION times a second!
As a result, power is only drawn from the battery about 1/2 the time, while the coil 'recycles' the excess power the other times. So the LEDs maintain full brightness without draining the battery continuously.
Step 4: Conclusion
Unlike series regulators (like the 7805 and LM317), which burns up extra power as heat, this circuit stores the excessive energy in a coil, to be re-cycled periodically, to maximize battery life.
I hope you will try and build one - as 'green' as LEDs are, their efficiency can still be improved by limiting the energy wasted in resistors and series regulators.
More information on this and other circuits can be found on my website.