So, you want to make a geocache that unlocks with a game of hangman?
Cool idea. How were you planning to power this, if it's a geocache? Batteries? What were you planning to use as the input device? A joystick arrangement hooked up to the LCD screen? I suppose it's do-able. Some LCD modules have tactile buttons on them - you could rig that up. Not sure about drawing the actual hangman, but a basic line drawing is probably possible on a two-row 5*7 LCD display.
Do you know how to program at all? Were you going to build the mechanical its and the enclosure, and just need someone to do the code? It's actually a fair bit of code, and the programming depends a bit on the actual LCD screen you want to use.
At the very least, you'll need some idea of how it is to look, what series of screens the use will be presented with.