First: Clean your piece of jewelry thoroughly with soap or Dawn liquid before you start. Scrub into the crevices. Body oils and lotions protect your skin by coating it. They will do the same thing here, preventing oxidation.
Believe it or not, this is generally the main reason that any oxidation process doesn't perform well.

Second:This process relies on heat to produce a chemical reaction between the metal and the sulphur fumes. If both the egg and the piece are hot, then the margin for success the first time goes WAY up!

Heat the jewelry you intend to oxidize while the egg boils. You can do this by placing it in scalding tap water.

I also use a slightly smaller container. After it is closed, I float it in the water that the egg was boiled in - again, more heat, more chances to get the result you're looking for the first time around!