I tried my hand at my own Mayo. I will not be eating it, because I am doing the Autoimmune thing. But my family will. And I put in my upcoming broccoli salad.
It may not be perfectly Paleo, but here is what I did.
2 whole Large Eggs
2 Tbs Vinegar
1/2 tsp Ground Mustard
1/4 tsp salt
1 1/2 cups of any combo Fat/Oil you like.
Pulse the first 4 ingredients in the food processor. Then poke a whole in a plastic cup with a toothpick (or larger depending on the thickness of your fat/oil), put the cup into the spout of your food processor, and fill with your oil. It will drizzle in slowly and eventually form a Mayo.
It worked nicely in the Broccoli Salad, and it may not be perfect, but it is better than store bought.