A garden-variety solar cell (2-volt, 20mA) is all that's needed, and the entire circuit consists of ten basic components.
Step 1: The Parts List
BC327 PNP transistor
BC337 NPN transistor
1N914 or 1N4148 Low power signal diode
100uF 6v capacitor
100uH inductor (A larger can-type is chosen for its lower internal resistance. Thanks to acmefixer for bringing this up!)
1 x Red LED 20mA
1 x White LED 100mA. A wide angle unit (90-degrees or more) is recommended here to avoid glare and hot-spots. I also chose the piranha style so I can get the 90-degree angle when I assembled it.
Connecting and Mounting hardware.
Step 2: Assembly
Allow the space but do not solder the transistors till the very end to avoid any heat damage to them.
Step 3: Circuit Operation
LED1 and LED2 serves 3 purposes. Firstly, they limit the output of the coil to about 4.6V so that the Lithium cannot overcharge. Secondly, the Red LED starts to glow when the battery reaches about 3.3v, and becomes brighter as we approach full charge. And finally, when J2 is turned on, LED2, a higher powered White LED is then turned on as the main light.
The output of the battery is so close to the Vf of the LED that we can get 90% efficiency.
The second image shows the faint glow of the Red LED against the sun as the device recharges.
Step 4: Putting together the lamp
The battery is a 3-volt CR2 Litium rechargeable which I wrote about here.