Mike T asks via Twitter,
One of the reasons you are finding a wide variation in instructions on seasoning cast iron is because it's a pretty forgiving process. Your goal is to develop a film of fat over your iron which will keep water and oxygen from reaching the metal and will consequently keep it safe from rusting. The layer also help keep food from sticking to the pan, making it a naturally non-stick surface.
In general, you need fat and some heat. You want a thin layer of fat on the pan, and a relatively high heat to do its science. The clever science trick behind a season on a cast iron pan is to cause the fat molecules to join together into a solid surface. Fat molecules join when they oxidize, and they oxidize especially well when they are heated up in an oxygen-rich environment like your oven. I mention the oxygen rich environment because, no matter how big your deep fryer, deep-frying your skillet is not going to season it well.
Fat can be divided into two major groups: saturated and unsaturated. Unsaturated fats have a great ability to bond with molecules, whereas a saturated fat is not able to bond easily with other molecules. Fats are composed of fatty acid molecules, and can be thought of as a barrel of monkeys. A barrel of monkeys is a toy with a bunch of plastic monkeys with hands, legs and tails that are curved around like hooks, so that when you get them together, they tend to link up. An unsaturated fatty acid would be a normal barrel of monkeys monkey. A saturated fatty acid would be a monkey that has its hands, feet, and tail straightened out. The unsaturated monkey will link up easily with all of the other moneys, where the saturated monkey is not at all interested in such things.
If you were to put a bunch of unsaturated monkeys in a barrel and shake them up, they would eventually become linked, and when you tried to pull them out, you would have a long chain or ball of monkeys. If you did the same with the saturated monkeys, some would probably link up a bit, but most would remain separate. This is essentially what happens when you heat up your fat in a cast iron pan: you are linking together the chains of fatty acids so that they become a solid mass.
So, what you want is a thin layer of an unsaturated fat, and the less saturated the better. The Mayo Clinic lists corn, cottonseed, flaxseed, grap seed, safflower, soybean, and sunflower oils as polyunsaturated oils, so I would probably lean towards corn oil for your seasoning needs. After all, with corn subsidies as they are, corn oil is cheap and plentiful.
Next, you want to ensure that the layer of oil is thin. You're effectively making a plastic coating over your pan, and you want it to fill the crevices easily and slowly, to ensure an even coating that builds up over time. The more patient you are with this, the better the end result should be. So pour a tablespoon or two of oil into your pan and swirl it around until it covers all of the inside surfaces. Feel free to add more as needed, and you might employ the services of a paper towel (or your fingers) to get the oil to some of the harder to reach spaces.
Heat your oven to 450°F, which is at or above the smoke point of refined corn oil (depending on your source). You want to hit the smoke point because you want your oil to start breaking down, which is what happens when it smokes, so that it is encouraged to form all of those bonds with the other fatty acids. The down side of this is that you're going to generate some oil smoke in your kitchen, so if it's a nice day, open some windows.
Place a jelly roll pan covered in aluminum foil on the bottom rack of your oven. Place the cast iron pan upside down on a rack above the jelly roll pan, ensuring that whatever will drip out will fall onto the aluminum foil. This will ensure that excess oil drips out, and aluminum foil ensures easy cleanup. If you like scrubbing and are concerned about the environmental impact of using extra aluminum foil, you may skip the foil, but if you skip the jelly roll pan entirely, your oven will start to smoke every time you hit 450 degrees until you clean it. Let the pan bake in the oven for two hours.
You will end up repeating this procedure a few times, and likely more over the life of the pan. Cast iron purists complain that newer cast iron pans are made with a rougher finish than older cast iron pans, which means that it is harder to make them nonstick. If you have an older pan, you will have a much easier time with seasoning than with a newer pan. Still, with care and effort, you can do a pretty good job, even with the rougher pans.
Never wash with detergent or scouring pads. What I do is, after cooking, I heat the pan back up just enough to melt whatever fat may be from the food, add some more oil if necessary, and pour in a 1/4 cup of kosher salt (I do not measure this - I just pour out some salt until it looks like enough to clean the pan). I use paper towels to scour the pan and remove any of the solid bits. Heat the pan after that until whatever thin layer of oil you added smokes, and let it cool. If need be, you can wash with a mild soap and water, but most of the time that shouldn't be necessary.
Oh, and an important reminder from Twitter that I neglected to mention:
@thefoodgeek I would add to dry the pan immediately with paper towels after washing and not letting it drip dry. Cuts down on rust.