Well, actually I already hacked http://user.baden-online.de/~pjanssen/radioaktivitaetsmessung.html (german site) an old russian geiger counter model (35 ? only ) and scripted a website upload function with python, but there were reliability issues with the software which ran on my work station, so I'd like to have a device which logs the data itself for some time even without the PC, but makes it available immediately but also backs it up if needed.
My hack is a simple interface which drives an opto coupler (so it can't fry my computer with it's some hundred volts needed for the geiger tube), and since I get the pulse from after a mono flop from the original device (point (3) between D1.1 and D1.2 if you look at the circuit diagram), it should be nicely formed.
So the arduino would only need to count these and back them up somehow if the PC (which will be a web server later) is not working, and interface to the PC of course.
Another option would be to add GPS so it could be used while driving in a car or walking around, and create space-time/radioctivity maps. In a way, it's for purposes of curiosity and exploration, who knows what you might discover, and if you keep it with you, you can track in space-time whatever values you find.
Count rate (http://user.baden-online.de/~pjanssen/daten.html, old values before my PC crashed, I didn't reinstall the software afterwards yet) of my setup is about 35 imp/min under normal conditions, but might be higher if things ever went bad (which never happened), or depending on certain other conditions like weather, height etc.
But options would be to attach several other tubes with higher rates (e.g. higher sensitivity). I know some people who arrange such tubes in a certain manner and also count coincidences in order to calculate a rate which doesn't take into account cosmic rays (when they pass through tubes arranged one over another and not horizontally). So this would need several counters, a completely new circuit anyway, and the pulses would probably need to be much shorter because of the higher rates, which the Arduino would need to be able to accomodate, but without loosing any pulses due to interrupts or software glitches or whatever, and I'm not sure if it is up to this task or if additional circuitry would be needed.
Maybe the arduino could also be used for regulation of the high voltage of a new circuit, although some other microcontroller might be better to save energy. After all, the geiger tube needs high voltage but needs only little current, and if the controller needs current in the order of some milliamps, it will be the biggest consumer, so here is where one should save battery life.