Get a pot of water to boiling. Once it's boiling, gently and carefully add the eggs into the pot. Set a timer for 12 minutes.
Once the timer goes off, drain the water out of the pot and add the eggs into an ice both (just fill a large bowl with lots of ice and cool water. Let the eggs sit in that ice bath for at least 30 minutes.
The key to peeling hardboiled eggs is that little thin membrane that is attached the egg shell. Gently tap the fat end of the egg to get it to crack.
Pick off the shell then look for that membrane.
Take your fingers and pick off some of that membrane. Now, continue to slide your fingers or the inside of a spoon under that membrane to lift the shell off. Sometimes it helps to roll the egg gently on the counter to get that shell to loosen up a bit. You can also peel it under some cool running water.
Once you have them all peeled, set them on a paper towel and gently dry them off.
Slice eggs in half lengthwise.
Carefully remove yolks (so you don't tear the egg whites) and add them to a bowl. Set aside the egg whites for now.
Mash yolks with mayonnaise.
Add relish, mustard, salt, and pepper; stir well.
Spoon yolk mixture into egg whites. You can also put the filling mixture into a piping bag or a ziploc bag with the tip cut off and pipe it into the eggs.
Sprinkle eggs lightly with a bit of paprika. Then enjoy! Keep eggs covered and refrigerated if not serving right away.
Notes
I always boil a couple of extra eggs in case I have issues peeling one or two. Save those mistakes and use them for a little egg salad sandwich. Or use the extra yolks to add to your filling so that you have nicely stuffed deviled eggs.
If you don't like sweet pickle relish, you can leave it out.