As described here, it is possible to add a Radiation Sensor Board to a Raspberry Pi by using

1. RASPBERRY PI TO ARDUINO SHIELDS CONNECTION BRIDGE
   
2. RADIATION SENSOR BOARD
   
3. A GEIGER TUBE
   

Would this work for the project -and if so, what Geiger tube do you prefer?

This project uses uniform designs sent periodically as kits, and being open source would support homemade versions using the same parts and source code, but does not support Raspberry Pi-connected tube shields like you propose (as far as I know). Members use the Pi here to run a BOINC client which is needed to collect the sensor kit's data and submit it to the server as well as perform computational work units, without requiring a desktop/laptop to be on all the time.

However homemade detectors of several types are supported by Radmon.org, with either serial, USB or audio (click only) connections. The SBM-20 tube is used in this project's kits and is a common tube used in many homemade sensors as well. Of the tubes available for your board, first decide if you want to detect Alpha radiation (stopped by paper, not likely to be found in background monitoring, but can be interesting), then decide on a size. The SBM-20 will not detect Alpha and does not fit neatly onto the board you chose, but is low cost.

The SBM-20 can be fitted onto the Raspberry:

It is just a matter of fitting instead of