The heart of the system is the Adafruit Ultimate GPS module. It requires 3 volt power and logic, so it will have its own 3.3 volt regulator, and the TX serial line will be level shifted with a diode and resistor. In order to insure that the GPS module will warm-start when powered on, the backup pin will be connected directly to the battery pack.
The system is powered by two 1.5 volt alkaline batteries. Whether you go with AA, C or D depends on how long you need the box to be able to operate. The battery input is pushed up to 5 volts by an NCP1402. That 5 volts directly feeds the controller, but is switched by the controller to power both the GPS module and LCD panel (both at the same time - if the display is on, then so is GPS), as well as separately turning power on to the servo output. The battery voltage is also fed to one of the analog pins on the controller so that it can check the battery state. When the battery dips down close to 2 volts, then we'll have to warn the user or perhaps take failsafe action of some sort.
The intent of the design is that the controller will power down the accessories and then put itself to sleep, meaning that the only power drain from the batteries during sleep will be the 5 volt power supply, the backup power for the GPS and the very small power drain of the sleeping controller. The button will interrupt the controller, waking it up. It will then power up and reinitialize the display and GPS. The GPS should be able to warm-start quickly (Adafruit claims it can do so in seconds, provided it hasn't been off too long). The controller will do... whatever the firmware programs it to do with regards to the GPS position and/or time. If it decides to unlock the box, then it will turn the display and GPS off and operate the servo (the 5 volt supply can only give 200 mA, which isn't enough for everything).
All the rest is just deciding what the rules of the game are and providing a UI.
At the moment, what I have in mind is an "open" and "closed" mode. When the box is in the open mode, there's an editor to set the game parameters (GPS location, tolerance and whatever else). There will also be a way to set a "cheat code." If you hold the button down when waking it up, it will prompt for the cheat code, which will immediately open the box and revert to the "opened" mode. This is needed if the batteries get low or something else goes wrong.
As with all the backpacks, it's an 80x36 mm board sandwiched on the back of a 2x16 LCD display module.
Unfortunately, it's going to wind up being a little bit expensive - the GPS module alone is $30, the LCD is $12, the board is $7 and the controller is $3. It's probably going to cost $75 or so when it hits the store.