This is a pretty basic oil lamp, but it works just fine, and the total material cost was zero. I have one that I fired up recently during a power outage. The basic elements are that you need a jar that will serve as an oil reservoir (and also possibly wind shield), a wick, and something to keep the wick protruding just above the oil. You also need oil. All of these can be had for free, just by diverting the materials from your garbage. In particular, the oil is taken from scrap that we might otherwise throw out. Photo quality here is low - the only camera I have access to at the moment is a cell phone camera.
Reservoir: While any container can work here, some will work better than others. Choose a glass jar so that light can shine through it, and with a wide base (so that it is resistant to tipping and can contain more oil). You will also want a lid for it. I built mine out of a pickle jar.
Oil: "Lamp oil" that you buy in stores is actually just very low-grade olive oil (sufficiently low grade that you can't eat it). You can use any other olive oil you have around, canola oil, any other vegetable oil... and a variety of other oils. You can also mix in some fats, so long as the overall mixture is liquid at whatever temperatures you're using it at. Using pure oil is a waste here. My oil lamp is filled by pouring the runoff from frying into the pickle jar, pouring in the leftover oil from tins of sardines/anchovies (and anything else packed in oil), and so forth. As a note, you'll rapidly see the value of the lid. The oil itself can have strong smells, especially if you include oil from fish tins. While burning it actually is quite odorless, but you will want to keep it covered when not in use. Below is a picture of my pickle jar, half full of waste oil. You probably can't tell from the photo, but the bottom has a fair bit of sediment - bits of sardine, little crunchy bits of fried whatever, and so forth. That stuff all sinks to the bottom and doesn't interfere with the operation of the lamp.
Wick: Ideally you want proper lamp wick here. Of course, oil lamps aren't really a tool for ideal times, and you can make do with a lot of other things here. I've had my oil lamp running fine with paper towel serving as wick. You can use any absorbent string, bit of rag, or the like. Just remember to let it soak up oil before you light it. When you see the 'lit' picture below, the wick I'm using is a bit of torn-up rag.
Wick Support: There are two main methods for this. You can use a bit of wire (that either rests on the bottom or hooks over the lid) and supports the wick. A bit of coat hanger works fine. Bend one end into a spiral that will hold whatever you're using as a wick, and then shape it into something that will hold it at the oil level. I tried this method, and I found it to be a giant hassle. As the lamp burns, the level of oil will slowly drop. You have to adjust either the level of oil (adding more oil or else dropping stones in to take up the volume) or the level of the wire. If you don't watch it, the oil level drops, the wick is no longer able to take up oil, and burns up. For a lamp you can operate more or less without watching it, you can make a floating wick. I made mine by cutting off the bottom of a pop can, folding down the sharp edges, and punching a hole in the top of the 'dome' at the bottom (see pictures). The wick is threaded into this and set into the oil. This picture shows the modified can bottom. Note that it is really nothing special - two minutes of work, and the application of some simple tool to make the hole.
The finished product has cost you absolutely nothing, and burns oil that you were probably going to throw away anyway. When I recently had a power outage, I fired up my lamp, which worked just fine and shed more than enough light for me to sit around and read by. It's honestly easy enough that you could throw one together in a pinch from things you have laying around. I just keep my pickle jar sitting in the kitchen to pour waste oil into. As a note, pouring waste oil into your kitchen drain is bad for your plumbing, so you do much better saving it for use as lamp oil. Below are pictures of the lamp in use, as taken with a crappy cell phone camera. The second picture is taken with it as the sole light source. Even just with my shoddy cell phone camera you can make out a fair amount of detail. It's no halogen floodlamp, but it'll certainly get you around the house.
