It really is easy
To make 1 cup of mayonnaise you will need:
1 cup of light olive oil (less strongly flavored than standard olive oil) or other good-quality oil, like walnut or sweet almond oil
1 egg
Juice of 1 lemon, or vinegar
A pinch of salt (and pepper, if desired)
Water to thin the mayonnaise
1). Separate the eggs in your recipe. Reserve the whites for other recipes.
2). Egg yolks contain a natural emulsifier, lecithin, which helps thicken sauces and bind ingredients.
3). Lemon juice or vinegar adds acidity to the mayonnaise. It also helps flavor the mayonnaise.
For each cup of mayonnaise, add between 1 and 2 tablespoons of fresh lemon juice or vinegar, depending upon your tastes.
4). Combine the egg and acid in the bowl, whisking to mix.
You can make mayonnaise in a food processor or by hand, with a mixing bowl and whisk.
The key for either method is to add oil very slowly, in a steady stream, while the processor is running or you're whisking vigorously.
(Stabilize a lightweight mixing bowl by setting it on a coiled kitchen towel.)
5. Continue to whisk constantly, adding the oil in a slow, steady stream.
If the mayonnaise starts looking too thick, add enough water to thin it to the consistency you desire.
Add about a teaspoon of water at a time.
When the oil is all mixed in, the mayonnaise should be thick and fluffy, with your whisk forming ribbons through the mixture.
If it never thickened and you're stirring a puddle, chances are you will need to start over. (Or, if you're still partway through the process, you can save the emulsion by adding another egg yolk, whisking vigorously. Add in remaining oil, plus extra for a double recipe.)
Adjust the seasoning with the salt and pepper and more acid, if desired.
Let me know how it turns out