• Bring the eggs to room temperature, and place them in a pot big enough so that you don't have more than a double layer.  Cover the eggs with cool water, cover the pan, and set over medium heat. 
• Bring to a boil, turn the heat down to a simmer, and cook for about 14 minutes.  Immediately move the eggs to a bowl of ice water for around 5 minutes. 
• Place the eggs back in the empty pot, put on the lid, and shake them around a bit to crack the shells all over.  Remove the eggshells - usually, they will peel easily.  Make sure to rinse them well, so that no bits of shell remain.
• Cut each egg in half the long way, carefully remove the yolks, and place the yolks in a small bowl.
• Mash the yolks very well - no lumps.
• Add to the bowl, the mayonnaise, mustard, olive brine, olives, salt, and pepper, and mix well.
• You can use a spoon to fill each egg half, or put all the filling in a ziplock bag, snip off a corner. and squeeze it into the egg halves. You can also use a piping bag with a fancy tip on it, which is how I did it in the pictures.

Variations

• You would probably not want to use all of the suggested variations in one batch, but pair up a couple of things - capers and dill for instance, chives and horseradish, or pickles and bacon. Whatever you think sounds good together. Keep in mind that the mustard will add a bit of heat, so be cautious with the horseradish and siracha sauce