Secret badges? Sign me up! :)
However, my take on it would be slightly different. Don't get me wrong, badges are cool and all. But another badge is just another number somewhere in a profile summary.
What users want are gravatar modifiers. That is the truly awesome spectacle of the hats. And the hats are awesome.
I would suggest that these secret badges, instead of being epicly hard to achieve and requiring a ton of time, are temporary. Temporary being in the time frame of 6 to 8 weeks (2 months, 6 possible per year). This time frame would allow for time to consider the next one.
Every 2 months (6 times a year) offer a secret achievement. This could carry a badge earning if you wished. Perhaps a silver badge that could be earned multiple times.
The achievement would vary, and so would the reward. The reward would be a gravatar modifier of some sort. However, at the end of the current achievement period (every two months), the modifier would be gone, and the only way to get the next one would be to complete the achievement.
The benefit from constantly coming up with new or different achievements would allow the SE staff to direct user traffic to the most needed parts of the exchange. And also, users would love being able to get a gravatar modifier.
They would have to be rather exceptional though. Such as hitting the review cap 10 days in a row.