Sounds plausible. I allready had that whole debate in my head because I slept badly so I'm putting my nose in other people's problem instead of minding my own.
So like, a simple pluviometer isn't much more than a beaker but then you have to go out there and write the results and by that point your feet might allready be wet. It should be fairly trivial to put a swimmer in said beaker to action something. Simplest I could come up with on my own terms without much research. Swimmer just serves as on/off contact for the pump. Primitive but a start. Said primitive variant could probably much be improved by playing with proportions, collect larger surface, steeper walls, smaller volume -> go for inch/min or 10min and then extrapolate. But there is more. Turn the swimmer into an analogue measuring device with the aforementioned tweaks, and some algorythmic magic you should be able to be a real engineer about it. Put like a little concrete syphon on the lowest point so the pump can start working early... Hehe if you're growing saffron, or magic beans, or the finest sinsemilla it might be worth the effort. I'm just going off in my head here.
I would be interest to know tho, how it performs vs say leaving the pump on overnight. Obviously it wouldn't be running as much, but it would be running much more often, whereas long term, some rains might drain, that were technically a bit too much, before you notice or act or do anything.