Place your egg yolk in a small, microwaveable bowl. (Save the egg white for scrambled eggs or another use.)
Whisk in the water, lemon juice and vinegar. Cover the bowl with wax paper and heat in the microwave for 25 seconds, or until the mixture starts to boil. Whisk the egg mixture again and heat for another 5 seconds, or until it starts to bubble again.
Whisk once more, cover, and chill the egg mixture in the refrigerator for 10 minutes.
After 10 minutes, whisk in your salt, basil and garlic powder. Then, very slowly pour the oil into your egg mixture in a thin stream, while whisking constantly. The mixture will emulsify and thicken into a creamy mayonnaise. (If you pour the oil in too fast, it will not emulsify.)
Store the mayo in a covered jar in the refrigerator for up to a week. See notes if your mayo separates into liquid.
Do not use extra-virgin olive oil, because the taste will be too strong. Also, homemade mayo can sometimes liquefy if the emulsion breaks. If that happens, place ½ tablespoon of water in a bowl and slowly whisk in the broken mayo in a thin stream, until the mayo comes together again.