This is a quick design I've done for a Rain Gauge that can be run from a micro controller such as an ESP8266. The parts uploaded here should be all the mechanical bits required. Adding a small magnet to the middle of the bucket would allow it to be read from a read-relay. Only thing that's left is to do is measure the volume in milliliters (or the unit of your choice) and then increment that rainfall count by that much every time the bucket flops over to the other side.

Note, I have not printed any part of this model yet, but I have validated that it should be printable. Would love to see if anyone prints this model. (I just have to many other projects on the go at the moment to spend more time on this right now).

There should be enough space inside to add an ESP8266 and a couple of batteries or run a cable from the read relay indoors. I left this bit of detail up to you. Just make sure your funnel/collector is waterproof enough to keep electronics underneath it dry.

Hope this is interesting and/or useful to someone. ;-)

Comments would be much appreciated.

Updated based on cbergthold's comments:
I've added Base v2.1.stl which raises the bucket a couple of millimeters higher, allowing it to tip a little more. It also now includes two spillways to get the water out of the base easier.
I've added Bucket v2.2.stl which is half the size to allow for more frequent tipping and to fit into the spillways.

update 2
I've made the spillways more printable so that supports are not required and also made the stands for the bucket a bit closer together (while still leaving space for a washer or two) so that the sensor can also be placed closer.

update 3
I added Outer Body sheet.dxf. The idea being that it would be quite a long print to print the outer body. These dimensions should allow you to cut out a flat piece of ABS (1.75-2mm thick) and then ABS weld the edges together to make the outer body in much less time that it would take to print this relatively simple shape.

Update 4.
Looks like the latest bucket does not gather enough water to tip it over. I'm working on an optimal design that needs the least amount of water while still being able to tip the bucket.

Update 5
Have a look at Bucket 2.5. I would apreciate feedback as I have not had time to print it.