The “old” way for an administrator to export a mailbox is by using Outlook. Basically, the administrator resets the password of the account then configures Outlook with the user’s mailbox that the administrator wants to export (or backup). Once Outlook synchronizes the mailbox’s content, it will create a data file.  

It is now possible to export a mailbox from Exchange Online with the Content Search feature on Office 365. This export only requires the eDiscovery permission (see the following screenshot).  

I’m going to describe how to export a mailbox with the new experience for Office 365. All you have to do is follow these steps: 

1. Go to the section Search & Investigation, then Content Search.  
2. New search.  
3. Enter the email address of the mailbox.  

4. Specify the location.  

5. Click on Save & Run, then name your query.  

At this point, you will see a preview of your Search result. Just go back to the Search section and you will spot your Search query name. Select your query and click on Export Results:

6. You will see multiple export choices, such as the following:  

7. Go to the Export tab to download the result of the previous steps (this action requires Internet Explorer or Edge). A tool has to be downloaded and launched.  

8. Copy and paste the key into the eDiscovery Export tool.  

You must choose the location, then you are done!