2006 piaggio BV 500 - intermittent speedometer/odometer. After the bike gets rained on, the speedo/odometer will always read zero, for a few days, then start working again. The saddle does not close completely due to the battery being the wrong size, so some rain might be getting in under the seat.

The water getting in is corroding the electrical contact from the speed sensor's wire, situated on the transmission. This can occur at both ends: either at the sensor or at the indicator.
You will have to find out which by looking at the sensor connector: If it's corroded then clean it thoroughly, crimp it a little more then insert it firmly and insulate it further with some hot glue. If the connector there is in good working order then follow the wiring up to the instrument panel; - check for wires with broken insulation and repair that. Finally if all the wires are correct then the fault is at the indicator: take it out, clean it and insulate both the connector and the instrument panel itself - the water might seep in and short the electronics board.