How can I add a custom button on GitHub.com which I could click on in order to deploy changes from the Master branch to a related Salesforce.com DEV Org?
What have you tried? Have you seen https://developer.salesforce.com/blogs/2020/01/using-salesforce-dx-with-github-actions ?
You can steal sample actions from SF official repo at https://github.com/trailheadapps/lwc-recipes.
You'll need to provide the secret login URL to your target org, there are blog posts how to generate it and store in GitHub variable, for example https://tigerfacesystems.com/blog/sfdx-continuous-integration/ or https://github.com/sfdx-actions/setup-sfdx
Last but not least - to have action available for manual run (not just automated) read up about "workflow_dispatch"