I have also been thinking about this. I had thought to give a arduino kit of some sort to family and friends for christmas but wanted to include an interesting project. This is by no means full answer but here are my current thoughts.
The project should be limited to use a sensor and control (motor). I'd like to have a project which can be applied in every day use. Hence, not something that teaches you the basics of controlling components, but something that tells you how to apply them. For example, a temp. sensor that logs the data to your computer so you can monitor energy efficiency of various parts of your house. Or, an IR sensor and motor to detect when your child gets home late at night. Rig a sensor so that you can monitor when your baby moves in their sleep, and log this. Audio sensor that detects when your partner snores and has a string attached to it which pulls their toe. Etc.