According to the Incredible Edible Egg people, (And thy are the eggxperts!) here's some tips on peeling an egg:
Peeling Removing the shell and membranes from a hard-boiled egg.
Opinion among researchers is divided as to whether or not salt in the cooking water helps make hard-boiled eggs easier to peel. Some research indicates that a 1 to 10% salt level (2 to 4 tablespoons per gallon of water) makes unoiled eggs easier to peel, but peelability of oiled eggs is not significantly affected. About 90% of the eggs available at retail are unoiled.
A nicely centered yolk makes very attractive deviled eggs and garnishes. However, as an egg ages, the white thins out which gives the yolk more opportunity to move about freely. This can result in a displaced yolk when you cook the egg. Using the freshest eggs possible will minimize this displacement, but very fresh eggs are more difficult to peel after hard boiling. Most researchers agree that using eggs that are “not too fresh” will help make peeling easier.
The air cell that forms between the shell membranes as the egg ages helps to separate shell from egg but, in very fresh eggs, the air cell is still small. The best compromise for attractive eggs with centered yolks that are relatively easy to peel seems to be using eggs that have been refrigerated for about a week to 10 days. Some new research suggests that yolk centering may be better if you store eggs small-end up for 24 hours before hard-boiling.
Immediately after cooking, thoroughly cool eggs in a bowl of ice or under running cold water; five minutes isn’t too long. Peel the eggs right after cooling for immediate use or refrigerate them in the shell in the carton for use within one week. To peel an egg, crackle the shell all over by gently tapping the egg on a table or countertop. Roll the egg between your hands to loosen the shell. Then peel off the shell, starting at the large end. Hold the egg under running water or dip it in water to make peeling easier.