To keep it within that cost AND be a built-from-scratch light, you might have to go with a 5mm LED over a power LED. Those are plenty for working in the dark. You'll have them making a Cub-Paklite, powered by a 9v battery. For real fun, add glow-powder on the epoxy so that you can find the things when they get set down.
I'm suggesting a 9v battery because voltage-boosting electronics are, at cheapest, just under $2 apiece. I'll talk about a voltage-boosting design later.
LED: You might go with
Super Bright LEDs (the 5mm ones from Sparkfun worked well for a project here) because these are cheap, and just give each Scout up to two for his light. If you want power LEDs, you get into needing heatsinking and stuff.
Get a pack of resistors
Buy a 9v battery connector for each. (Radio shack has these, at least)
Get a little slider switch for each - a "Single Pole Double Throw" gives 2 settings (and Off). Rocker switches are easier to waterproof.
Get some of those LEDs.
Each scout will be making a 2-speed light by choosing 1 or 2 LEDs, and a resistor value for each mode. To calculate how bright the LEDs will be, you need to know the current. That's calculated like this:
Battery voltage = (all LED Forward Voltages) + Resistor Voltage
9v = (about 3.2v per LED) + (resistor value) * current in amps
9v - 2*3.2v = resistor*current
A 5 ohm resistor would give
2.6v = 5*current
current = .52 amps
For a 5mm LED this is way too high - you want them down at 0.03 amps (30 milli-amps) - resistor values around 90 ohms are good for 'high,' and around 100 times that for 'low.'
Once wired, you can put epoxy on the wires, 9v plug, LEDs, and switch body. Glow powder in the epoxy will charge from sunlight and the LEDs. Once this is stuck on a 9v battery it'll run for a looong time. The 9v connection might rust, but getting wet won't hurt it too much as water isn't usually very conductive. A slide switch will self-clean as long as it doesn't get stuck.
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Power LED flashlight.
Needs power LED
Voltage-boosting driver
Heatsinking
Switch
A reflector or optic is nice
For a power LED light you'll probably want a voltage-boosting circuit. This is one I've used in a helmet light:
I have in mind for you. It won't deliver a lot of current, but that's ok - 200 mA is plenty for a flashlight. This one is single-mode, but there are other options, like You wire the outside ring to battery (-), perhaps the flashlight tube; and the nub on the front of the battery goes to the middle circle.
With power LEDs comes the need for heatsinking. If you can't get metal bodies on the lights donated or cheap from some local metal shop, you can always use a half-inch copper pipe end cap for a metal head. This'll work well with a PVC battery tube, although I'm not sure how well you can bond the two together. You'll need to drill one or two holes in the end cap for the LED's wires. Thermal epoxy will make the LED give its heat to the copper, and the end cap will hold a reflector or optic nicely.
You'll need a place for the LED to sit, a spot for the driver (The top of the battery tube near the LED works best), and a way to hold the battery and switch the driver off and on. Never connect an LED to a powered driver.
I'll add some diagrams so you can see what I'm talking about.
Drivers: Look for ones like these (DX used because they have many things, not because they are good). Cheap LED drivers will usually have more power with higher voltage. That also means that as the batteries die, the lights will get dimmer and draw less power, so it lasts a bit longer. Also, adding a resistor between the driver and the battery makes a handy 'low' mode on single-mode drivers.
[url=http://www.dealextreme.com/details.dx/sku.4451]1.5v, 300 mA single-mode
1.5-4.2v ~200 mA single-mode
1.5-4.2v ~600 mA single-mode