DIY schematic for solar lamp
You want to look for "solar lamp schematic", it will give you many schematics for making one yourself on google.

Here is one  from http://www.talkingelectronics.com/projects/SolarLight/SolarLight.html which explains how and why it works, as well as having some variations on it.

The reason most people just use a dollar store solar light is because it's insanely cheaper then getting the parts yourself. Just buying the solar panel itself will cost you 5 bucks at radioshack, or a few bucks on ebay plus shipping time. Then you need the rechargeable battery, and the inductors. It's not very cost effective at this point.

But in essence, you only need a few parts. The solar cell which charges the battery during the day, the battery which runs the led at night, and a small control circuit to prevent the led from lighting up during the day.