Let all of the ingredients reach room temperature before combining. This is crucial, cold ingredients can inhibit the emulsion process.
Add the egg to the cup that came with the immersion blender and blend for 15 seconds. Alternatively, you can use a bowl that is not too large. The mixture needs to make sufficient contact with the blade to emulsify.
Add the dijon, vinegar, and salt and blend for another 15 seconds.
If you want a more boldly flavorful mayo, substitute 1/4 cup of the neutral oil for olive oil.
Very slowly (drops at a time) add in the oil while blending with the immersion blender. Don't pour too fast or you could be left with a liquidy mix that failed to emulsify. Once about 1/4 of the oil has been added, you can start to pour in the oil in a small stream, while continuing to run the blender, until all of the oil has been added.
Once all of the oil has been added, scrape the edges of the bowl and mix again for another 15 seconds. Add any additional lemon juice, vinegar, or salt to taste.