This is the method I learned as a child from my mother and my grandmother. Proper oral tradition and all :-) There may be better or more efficient ways to make mayonnaise, but this one works and it's the one I know... Also, as you can see from the ingredients, this is the Mediterranean version rather than the Anglo-Saxon one (which seems to involve vinegar, mustard and no olive oil).

Ingredients:
1 egg yolk
about 150 cc of oil
salt
fresh lemon juice

Tools:
1 container for making mayo (aka mug)
1 cloth
1 stirring tool (aka tool)
1 bottle or similar container for pouring oil; a spout is useful (aka bottle)

Notes on ingredients and tools:
- The egg yolk should be at room temperature. If you keep your eggs in the fridge (I don't), take it out a good while in advance. It really helps.
- The oil can be anything, but like my mom+gran I generally use half olive and half sunflower or other flavourless veg oil. It can be done with all olive oil, especially if you have mild olive oil: I prefer strong olive oils so that's what I have in the house, and that's why I use a mix. I wouldn't recommend veg oil only, the taste of olive is important.
- The container: I used a mug, which is the perfect shape, but next time I'm probably going to use the graduated plastic jug (because of the stirring tool, see below). Apparently the shape is relevant, i.e. it should be relatively narrow; and if it's too tall, stirring will be more difficult. Or it may not matter, but this is the way I've been taught...
- The stirring tool: I used a teaspoon. However, it got really greasy (surprise surprise), and it was really hard to grip, and I got more cramps than I needed. A longer tool with a wider handle would be better. My mother always uses a fork, but I don't because I hate even the thought of the prongs scratching the bottom of the mug. So next time I'm going to try a soft-plastic container, and a fork.
- The cloth is for holding the mug, because you'll need both hand to pour the oil while stirring. A barely-wet J-cloth worked perfectly.

The method
Take the tools, the ingredients, and a lot of patience. Sit comfortably at a table. You can prop a book open and read it while you do this :-). Reverse the instructions if you're left-handed.

Put the cloth on the table, put the mug on the cloth and the egg yolk in the mug.

Start stirring the egg with the tool in your right hand. It doesn't need to be fast, but it needs to be steady. Faster stirring may make mayo faster, but slow works just as well, and you have to keep it up for a while, so on balance slow tends to be better. Steady is really really important: in fact, steady is the whole trick. And never never ever reverse the direction of stirring.

Start dripping the oil in with your left hand, really drop by drop, stirring nice and steady all the time. Keep going. Don't stop.

Once the thing starts looking and feeling somewhat like mayonnaise, you can start pouring a bit more oil at a time, but moderation is still good. At this point, you can also stop stirring briefly if e.g. the children are on fire or your hand really hurts, but don't overdo it. Keep stirring steadily and pouring oil until the desired consistency is reached. There may be some oil left over -- the mixed oil is quite nice for stir-frying or shallow frying, or even for salad.

Last night, this took me about 15-20 minutes of tireless stirring. It can probably be done in 10 minutes with practice (my mother is certainly faster), or stirring a bit faster; but I wanted to be on the safe side and I think that's the best thing to do until you get the feel of it. I used to have the feel of it when I was a child, but last night was the first time I made mayonnaise in 20 years or so.

Last, add the lemon juice and salt, and stir again. My mother said the juice of 1/2 lemon, but I think that's really too much. I used a couple of teaspoons and it was ok. It's up to your taste anyway. Same for salt.

Voilà!

What can go wrong:
The mayonnaise can go crazy. No, seriously -- that's the Italian word for it. Impazzisce. Apparently the English word is "separate", which is boring.
This happens mostly if you aren't steady enough, stop too long, or pour in too much oil at a time. Unfortunately, I don't know how to rescue a crazy mayonnaise. Well, the traditional trick is to put in (part of) a boiled potato -- I suppose the starch will pull it together, but I've never actually tried it so I can't say if it works. Luckily, if you're patient and steady it doesn't tend to go crazy in the first place, which is the best solution.