An important point in making mayonnaise is to use ingredients at room temperature. If they are cold, the sauce is more likely to split. I get my egg or eggs out of the fridge and put it/them in lukewarm water for 10 minutes or so. But I got overenthusiastic the other day, using water that was too warm. The combined egg and oil did not thicken, merely producing a runny liquid.
I hesitate to say this, because it is asking for trouble: this is the first mayonnaise ever to have gone wrong on me. I have never had one split. I should add that I do not make mayonnaise very often; perhaps for that reason, I am cautious -- usually -- about the temperature of the ingredients, and about the speed with which I add the oil. I always use mustard, which assists in the amalgamation.
I use a pestle and mortar. The heavy mortar stays steady while I work away. But I have also made mayonnaise successfully in a bowl with a small whisk.
1 egg yolk
1/3 tsp Dijon mustard
75 g olive oil and 75 g groundnut or sunflower oil, combined
Combine the egg, mustard and a few grindings of salt in a mortar. Add the oil a drop or two at a time, stirring vigorously with the pestle to amalgamate it before adding any more. When the mixture gets very stiff, thin it a little with a drop or two of lemon juice -- I keep it to hand in a saucer, and add it with a teaspoon. Continue to add the oil, carefully; once you have used about half of it, you can start pouring it in more generous amounts. Use the lemon juice to bring the mayonnaise to your preferred consistency. I like mine wobbly.