I'm leaving the mechanical design to my brother, but I will stress the need to minimize mechanical load on the servo.
My own recommendation to him will be to place a mechanical latch on the outside of the box to take incidental mechanical handling stress and only use the servo to prevent intentional attempts to open the box (and, again, this is only a game, not a true security application - it'll undoubtedly be easy to destroy the whole thing and get in the box if you wanted to).
The NCP1402 can do 200 mA. My design has a power transistor to control power to the LCD and GPS module. The plan is to turn both off before engaging the servo. With no load on the servo, and with the ATMega having a reasonably wide voltage tolerance, it ought to be able to make a go of it.