For lubricating bicycle chains, and other parts, we need a light machine oil, perhaps thinned a little bit by a solvent to help it get into the chain. Furthermore, an anti-rust additive would be helpful.
I found an oil which meets these requirements and is inexpensive (paid about $5.50 CAD before taxes for a 500 ml bottle at Vancouver Auto Parts). On the bike chain, it doesn't pick up a lot of dirt and isn't difficult to remove with detergents. It happens to be an air tool oil.
Not every light machine oil will necessarily have such anti-rust additive. Why it is important in an air tool oil is that these tools are exposed to water which condenses in the air compressor and makes its way into the supply lines.
This is the specific product I'm using:
The ingredients are published in its MSDS (material safety data sheet) accessible from the above product page. The exact formula isn't given away, but the MSDS gives us a good enough idea about what is in there. It's mostly hydrotreated oil (hydrotreating not only removes sulfur, but improves the quality of oils: the Wikipedia page does not tell all), a bit of napthtenic oil (thinner-bodied class of oils with some useful properties), a hydrocarbon distillate to help the product act as a penetrant, and a zinc-based anti-wear, anti-rust additive.