Push notifications with minimal data usage

In my project, I want to call an action on a Raspberry Pi from the Internet. Somehow like that:

1. Visit webpage
2. Hit a button on the webpage
3. Raspberry pi executes a script

The difficult part is, that the raspberry pi only has a mobile Internet connection without a flatrate. So, sending and receiving data from/to the Raspberry Pi is costly.

So this is my question:

Which technologies and patterns can I use to send push notifications to the raspberry pi with minimal data usage?