Nowadays, home irrigation systems may be equipped with one of a number of different devices wired into the controller.

Perhaps the most crude is mount a small rain gauge which shuts off the system.

More technical, also costly systems, use tensionmeters in the soil, again wired to the controller.

And it's very likely you can install a system which texts/phones/emails you.

The rain gauge that shuts off the system can be set to different rain accumulations. The system will remain off until the rain gauge has dried sufficiently. The drying rate of the rain gauge is supposed to mimic sod. So if you set your rain gauge to one 1" of rain your system will not run after the rain amount has accumulated to 1" and will run again after there has been sufficient evaporation, so it would depend on humidity and heat.

https://hackaday.io/project/12651-rain-water-irragation-controler

I am working on my own system to do just this I plan to add a tipping bucket rain sensor as well so I can adjust the amount of water to add after a rain fall and maybe delay a watering if there is rain forecast in the next couple of hours

There is also Wise Orchard's method: we get real-time information from local weather stations near you to adjust your irrigation according to local weather conditions, including rainfall. See our website for more info.

Yes, 'smart' systems are usually overkill, and not cost effective for most residential systems. A simple rain sensor, with cork absorbers, correctly set up, is enough to break your common and will interrupt your irrigation signal until it dries out enough to allow the signal again. We use the Irritrol models both wired and wireless. A word of warning...materials purchased from a retail store/box store are not usually the same quality and do not perform like those used by professional Licensed Irrigation Contractors.