Hack A Day has a nice DIY project tutorial on making a simple digital synthesizer.
The final result is not much of a synthesizer – but it’s cheap and it makes some noise:
This simple guide will show you how to build a digital synthesizer that generates and manipulates square waves. Your synthesizer will have one oscillator, which produces a variable pitch controlled by a potentiometer, as well as an LFO which modulates that pitch at a variable frequency. The part count for this project is quite low, and it can be built for under $20.
The project does a great job of letting you know where to get parts and how much they will cost. What’s cool about projects like this is that, once you figure out how they work, you can easily adapt them to create all sorts of bizarre musical devices.