Answer:
This information is stored in the Google Cloud Platform console. You can change this by editing the OAuth consent screen of your project.
More Information:
When you click the Enable the Drive API button on the Quickstart page, what is happening in the background is a GCP project is being created with the name Quickstart. This name is what is given to your project when registering it so that it can be set up for users to authorise. As it is a demo 'quickstart' project, this is the default name that it is given.
Changing the Project Name:
You can see all your GCP projects by visiting the Google cloud platform console. Here, you can create, delete, and change the settings of projects that you have.
At the top of the page, on the blue header bar, there will be a dropdown menu next to the Google Cloud Platform title. Clicking this will open up a list of your GCP Projects - though if this is your first time using the APIs you will have only one, the one called Quickstart.
To change the name, you will need to open the hamburger menu on the top left and follow ≡ > APIs & Services > OAuth consent screen. Here, you can click on EDIT APP next to the Quickstart title and change the application name to whatever you like.
There are other things on this page you can also edit, but if this is your first time using the G Suite APIs and/or the OAuth flow, I recommend reading up to get an overview of how the authentication process works and why this is important.
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