The desired key-feature for me is the ability to register functions of objects to the JSON-RPC server task, which then calls the function, in case the respective JSON-RPC-call was received.
How can this be done?
I found https://github.com/cinemast/libjson-rpc-cpp , but the concept of the stub-generator looks a bit overdone to me.
libjson-rpc-cpp does not require you to use the stub generator. You can always create the bindings manually.
Additional C++ frameworks for JSON-RPC can be found here.
But since you asked for an embedded compatible framework, I'd suggest you stick with libjson-rpc-cpp, since its the most lightweight framework, where you can disable certain features during compilation.
It is also worth mentioning, that I am the author for said framework.