Have you easy access to the rain gauge?
I would recommend opening the gauge up to see if there is anything obviously wrong with the tipping mechanism. I've have insects build nests around the tipping part and keep it from moving.
If it is hard to get access to you might try pouring some water in the top slowly and listen to see if you can hear it tipping with a sort of a thunk each time it rocks back and forth.
If you can get to it and open it easily and there is no obvious insect nest or such, (and look for a broken wire, etc, all of which I've had), the next thing I'd suggest is shorting the two wires together. I have one where the wires attach to a solder point and is easy to do with a short piece of wire, and others where the switch is well insulated. If you can try touching the two wires together and opening them back up with a jumper wire, again slowly perhaps once every ten seconds, and see if the count is seen on the console.
That would be a start. You could have a bad switch that doesn't close the wire connection when it tips, and you might have bad connections where the wires plug into the junction box as they arrive from the sensors and where the final wire goes to the console.
