Action delegate doesn't show in inspector.

Unity comes with components that have references to methods assignable through inspector (such as http://docs.unity3d.com/ScriptReference/UI.Button.html, the on click callbacks from gui buttons).

Yet I can’t figure out how to add similar functionality to my own components.

I’d say what I’ve tried so far, but unfortunately I haven’t found anything to try yet. I’m hoping there’s a simple solution to this.

You’ll need to use UnityEvents so that you can have them in the inspector.

Take a look:

You might also want to see this video: