I have an old Soviet personal radiation dosimeter that I broke as a kid by dropping it into water. I opened it up a while ago and the electronics in there were quite simple - mainly a ~10cm Geiger tube and what looked like Soviet variants of standard logic ICs. The tube looked intact - is there a way for me to test if it's worth salvaging, though?
You have to apply a high voltage, like +500 V DC via 3 or more series resistors to the central anode, and connect the cathode via a 100 k\$\Omega\$ resistor to ground. The HV supply has only to source less than a mA. If an ionizing particle passes through the GM-tube, it will cause a brief discharge current, which in turn will give a brief pulse over the resistor.
You'll need a radioactive source to test it. The higher the radioactivity the more pulses per minute you'll get (and the more unhealthy it is!). The pulses are short, IIRC in the order of microseconds, depending on the kind of particle. You'll need a storage scope to see them, or trigger a monostable multivibrator (MMV) with them, so that they're stretched in time.
You can quickly hookup a high voltage source from 555 timer, mosfet and transformer, then do everything as described by steventh. You can connect a piezo crystal to output - this way you'll hear clicks when particles enter tube.
You can get away without radioactive material, if tube is sensitive enough, you'll be able to detect background radiation with it.