Definitely a little cheese (a strong, good melting cheese, perhaps a reblochon or similar): cold meats and/or wild mushrooms are a welcome addition.

What I really care about, though, is the preparation. For me, an omelette itself should be nothing but egg, salt and pepper, and should never be flipped, allowing it to be slightly runny on one side. The cheese should be added as the first filling, to bind with the runnier egg, and the remaining fillings added later to warm, but after the egg is cooked enough so that they do not mix with the egg. The omelette should then be served folded, enclosing the filling.

To me, that is a 'proper' omelette. Good, free range eggs are a necessity too!