There aren't too many action or shoot'em up games that you can create with a 16 x 2 LCD module, however reddit user "rjk79" has produced a quite playable version of "Space Invaders". The display is effective, and the key to playability is in the controls - in this case the use of a Nintendo Wii nunchuk. These can be easily interfaced with an Arduino as they use the I2C bus and there's a great library available. Thus the game is a success, for example:
If you're interested in more advanced Arduino projects and usage - get yourself a copy of "Practical Arduino" by Jonathan Oxer and Hugh Blemings:
Create your own Arduino-based designs, gain an in-depth knowledge of the architecture of Arduino, and learn the easy-to-use Arduino language all in the context of practical projects that you can build yourself at home. Get hands-on experience using a variety of projects and recipes for everything from home automation to test equipment. For more information and to order, click here.