What you'll need:
A Geiger counter with pulse output (I'm using the Gieger counter from MightyOhm)
A few strings of battery-operated LED decorations (I'm actually using USB-powered strings of stars from Poundland - you want something that'll comfortably run off 5V)
Arduino-based microcontroller (I'm using an UNO)
Female header (three sockets' worth)
Male header (I used an 8 pin run)
5V power supply for your Arduino (this will also power the Geiger counter)
Assortment of connecting wires (I used a combination of multi-core, ribbon cable and solid core)
Soldering equipment, wire cutters, wire strippers etc.
Insulation tape or heat shrink
Arduino IDE and an USB cable to connect to your microcontroller
A jar or other receptacle to put the LEDs in
I'm also assuming you've read the original instructable - if not, now might be a good time to do so!