To some extent, shell hardness is out of your control: it depends on the chicken's diet.
The chickens must be fed with calcium-rich food and have plenty of exercise. (1)
Free range or farm eggs have harder shells because of the better diet that the chickens have. (2)
Yard chickens lay ... harder eggs in the early spring compared with some other times of the year.... Supermarket eggs are always less hard than the smaller, brown, country eggs and the citizens of Marksville would not be caught dead on the courthouse steps with anything but yard eggs.... regardless of the breed, the hardest eggs are produced by well fed, active chickens. It is particularly important that the hens get adequate calcium. (3)
If you have the option to buy your own egg, that is likely to give you a leg up before you even start preparing.
In addition to finding a well-fed chicken's eggs, or even if you're stuck with supermarket eggs, you may be able to select a hardest specimen out of a carton. This is something that probably takes a lot of practice, though!
The method generally used by the serious knockers for finding those hard eggs is to lightly tap them on their front teeth. According to [one serious egg tapper], the harder eggs will make a light high pitched ping, while the softer eggs will make a blunt, dull sound. (3)
The shell is naturally thicker at the narrower pointy end, so use that end for tapping -- and try to make sure you've got good support behind it. Boil the egg very well, and keep the air pocket at the wide end of the egg.
Proper boiling of the contest eggs is also a serious issue. Some rules are well known, such as eggs must be boiled tip down, so that the air pocket is on the butt end. (1)
They are boiled slowly, so that they will not jump around and hit the sides of the pan or other eggs.... the eggs must also be boiled point down. This is to insure that the air pocket ... will not be at the small end. There must be something solid behind the hard shell in order to keep it from cracking quickly.... [to keep the eggs oriented during boiling] both Brent and Mike actually boil their best eggs inside a cardboard carton. (3)
There's also a tradition of boiling in coffee grounds, but I have no idea how this would possibly work -- it may just be that somebody did it once, and also happened to have naturally harder eggshells, and so an old wive's tale was born. But I guess it can't hurt.
The old-timers believed that boiling the eggs in coffee grounds made them stronger. Some people still do it. As Judy Bordelon, Mike's wife said, "We boil our best eggs in coffee grounds, just in case...." (3)
Possibly cheating, and certainly not culinary, ideas
While the following suggestions will probably keep the egg edible, they are a little more likely to be visibly apparent -- Hey, why is Marc's egg so shiny?!? -- and also may result in resentment among the competitors, or outright disqualification, depending on the seriousness of your family. (Or, they may be delighted by your creativity!)
- Dipping the egg in sodium silicate (4)
- Apply a thin coating of epoxy or glue (e.g. ModPodge) to the outside
- How to Win at Greek Egg Tapping
- If Your Eggs Are Cracked, Please Step Down: Easter Egg Knocking in Marksville
- Hardening the shell of an egg