Preparations

[Uthgaard]

A crank generator is probably the simplest thing to create. The most complicated part would be deciding what kind of battery setup could accommodate that blends what you can afford with performance attributes.

Anything that creates a difference in electric potentials creates a voltage, you can think of it like water pressure. Rechargable batteries restore some of that charge. Practically, batteries in this situation would perform best when they aren't at a full charge. Most of the time spent charging a battery is when the battery is almost full. It takes less than half as much energy to charge a battery to 75% than it does to charge it the last 25%, so it's a lot like a concentration gradient made up of ions instead of solute.

The tricky part comes with spending the energy. You could create a battery pack out of batteries connected in series to charge them, and then just remove batteries to match the voltage requirements of whatever you're trying to power, but that's impractical for all but a few examples. Realistically you'd just need a device between the power supply and its final output that can regulate the voltage and current so that your device a) has enough to function and b) doesn't fry it. It's often a pretty thin line to walk.

You aren't likely to manufacture an LED. Diodes are essentially miniaturized vacuum tubes. You would have a lot of difficulty (read, basically impossible to DIY) growing the crystals to the required specs and without flaws, or to create the diode even if you did. On the other hand, they're really cheap since the manufacture is simple on the large scale in a specialized facility. Better off buying what you need.