What project is good for a total beginner
I would suggest grabbing a microcontroller starter kit of some sort - whilst running through the basic tutorials you will probably get many ideas of things you would like to try (plus have a little better idea of what may be involved)  

The Arduino platform is any easy introduction into the world of microcontrollers, but if you want a real challenge it maybe better to start directly with the native tools.  

Microchip make some good starter kits - something like the PICKit3 Debug Express comes with a small 18F demo board and programmer/debugger that can be used for future projects.
If you would like a more powerful microcontroller, then the 32-bit PIC32 starter kit is well worth checking out.  

For projects consider things like (when you get past the initial tutorials):
A digital clock, small line following robot, frequency counter using CCP module/comparator and display on LCD, simple tone generation, movement (e.g. human presence) detector, LED matrix/cube, IR remote control, serial communications with other micros/PC (SPI/UART/I2C)   

Good luck!