Tempered oil (oil at around 70 degrees F.), if needed
Egg yolks, if needed
Warm water bath, if needed
Hot water, if needed
Step1
If your mayonnaise was made with ingredients not of the same temperature, the mixture can sometimes break. The egg yolks and the oil (two main ingredients) should be 'tempered', or brought to the proper temperature for the recipe, to about 70 degrees. If the ingredients are too warm, the fats can break down easily. If they are too cold, they will not emulsify properly. Bring the broken mayonnaise to room temperature. Break one yolk into a mixing bowl, and whisk in the broken mayonnaise tablespoon by tablespoon until the mixture is cohesive, then add in a bit of the tempered vegetable or olive oil, or whichever oil you used for the recipe, to set the mayonnaise.
Step2
Another reason mayonnaise made in house breaks is the oil and yolks were not properly mixed to start with. After the egg yolks and flavoring ingredients are mixed together (the mustard, salt, pepper, paprika, etc,) the oil must be slowly added to the yolk mixture to produce a proper emulsion. If all the oil is just dumped into the yolk mixture at the start, it will be nearly impossible to bring it together again. But if this happens, take a yolk or two (depending on the amount of mayonnaise to begin with) and beat well with a whisk in a small mixing bowl. Then, while whisking the broken mayonnaise, add in a very steady, slow stream the beaten yolk(s) to bring together.
Step3
Sometimes if a mayonnaise is made ahead of time and stored in the cooler for prep earlier in the day or the day before, the mayonnaise can separate if the temp of the cooler was too cold. What can help this is to use a bain marie with warm water. You can make your own by bringing out a large bowl (larger than the one holding the mayonnaise) and add hot water, but not boiling. Hold the smaller bowl with the cold, broken mayonnaise over and slightly in the water to warm up and whisk until the cold mayonnaise comes to temp and is brought together.
Step4
The last step may fix a broken mayonnaise if you don't know how it broke in the first place. If it is not pure liquid (adding just too much oil to the recipe in the first place) then the mayonnaise may have just separated due to temperature fluctuations (in the ingredients, making or storing) and sometimes all that is needed is to add in some hot water. Add 1 teaspoon at a time to the broken mayonnaise while whisking. The mayonnaise will usually come together.
Tips & Warnings
Remember that a homemade mayonnaise contains RAW ingredients, and raw eggs may be contaminated with salmonella. Always use common sense when making homemade mayonnaise, and never serve it to children or people who's systems cannot handle unprocessed or uncooked foods.