The Inspector is part of the Babylon.js repository and can be customized in the same way as Babylon.js can.
Before following any of the steps below knowing how to customizing Babylon.js in general is required: How to Start Contributing to Babylon.js
As all the code of the Inspector lies in Babylon.js repository, you have first to follow this tutorial to setting up your IDE.
In a terminal, type :
cd Tools/Gulp npm install gulp inspector
Once this is done, run the test task (CTRL + SHIFT + P) and type
test then press enter.
Open the url : http://localhost:1338/inspector/index.html You should be able to see the test file (see next) for the inspector.
The test file is:
/inspector/test/index.js (here in Github)
You can update the test file, and press F5 in your browser to check the changes.
Babylon.js Inspector is based on a React. You may want to read more about React before updating the inspector: https://reactjs.org/