eclectica has posted a circuit for a PICAXE-enabled raincounter. The device employs a 1mm/pulse rain gauge to measure rainfall. Pulses from the gauge’s internal reed switch are read by the PICAXE-08 MCU, which in turn stores the running count in EEPROM and displays the resulting measurement on a serial LCD readout.
While the exact type of rain gauge employed in the project is not indicated, an example is found in this 22 page PDF datasheet for a Tipping Bucket Rangauge from Campbell Scientific, Ltd.
The project description at Circuitlab contains the schematic and Picaxe code.