The traditional way is the following (from the top of my head)
Bring all ingredients to room temperature before proceeding. Also, on humid days it will be more difficult to make mayonnaise by hand. (But with a blender or rotary mixer you should be fine.)
1 scant cup oil (use a lighter olive oil, or 1/2 extra virgin 1/2 any neutral oil, as samclem’s suggestion.)
1 large egg yolk
1 -2 tsp vinegar (use white wine vinegar) or lemon juice
Dijon mustard
salt and pepper
Traditionally, this is done by hand. It is quite tiring, but the results are worth it. Otherwise, use a handheld rotary beater. I don’t like the texture of blender mayonnaise. Homemade mayonnaise is much silkier than its commericial cousins.
Begin by beating the egg yolk, vinegar (or lemon juice), and salt and pepper together in a bowl until it becomes a bit frothy. At this point you can start adding your oil. Pay strict attention to the following: The oil must be add, at first, very slowly, drop by drop. Make sure the oil is absorbed by the egg yolk before adding the next few drops. You shouldn’t be mixing in more than about half a teaspoon a second. Eventually, your mixture will begin to thicken. After you’re 1/3-1/2 way through your oil, you can start beating it in with a slow and steady stream. As before, make sure the oil is thoroughly blended in before adding more.
Then you should be finished. Your end result should look like a shinier and silkier, but thick, version of commercial mayo. At this point you can add Dijon mustard to taste. This mayonnaise will last you two days at most.
Hints:
If at some point your mayonnaise “breaks,” ie separates back into yolk and oil, you can usually save it. In another bowl, put two tablespoons of hot water and beat the broken mixture slowly into this bowl. It should re-emulsify before your eyes.
Mayonnaise sounds tricky, but it’s not that difficult. It just takes patience in making sure each drop of oil is absorbed fully before adding the next one. I urge you to use a rotary beater rather than a blender. You will get much nicer results that way.