Rather than just toss it, I wanted to see if I could remake it successfully. I did!
It was very simple. I used one egg yolk and the oil mixture from the failed mayonnaise. I left them both sit at room temperature for an hour or so. It is critical that they both be at the same temperature.
Then I simply whipped the yolk by hand with a whisk. Gradually, and I mean gradually, add the oil mixture. I started with about a half a teaspoon and very gradually increased the amount of oil when the mixture was emulsified. Eventually I was able to add by tablespoonfuls but I did not take a chance on a quantity larger than this.
I will also share my recipe. However, I am including the instructions for making it in a blender or food processor. Alternatively, I would make it by hand with a whisk.
2 egg yolks
3/4 cup olive oil
3/4 cup canola oil
1/2 teaspoon of mustard powder
3/4 teaspoon of salt
pinch of cayenne pepper
4 to 5 teaspoons of lemon juice
Place yolks, salt, mustard, sugar, pepper, and 3 teaspoons lemon juice in blender cup or work bowl of a food processor fitted with the metal chopping blade, and buzz 15 seconds (use low blender speed). Now, with motor running, slowly drizzle in 1/4 cup oil (use moderately high blender speed). As mixture begins to thicken, continue adding oil in a fine steady stream, alternating with remaining lemon juice. Stop motor and scrape mixture down from sides of blender cup or work bowl as needed.