We can use the Exchange Management Shell to remove the default discovery mailbox, recreate it, and then assign permissions to it.
Why do you want to do this?
In Exchange Server 2013 and Exchange Online, the maximum size for a default discovery mailbox is GB. It is used to store in-place ediscovery search results. Organizations can increase the storage quota to more than a few gigabytes before the size limit changes. Therefore, the discovery mailbox can be increased to more than a few gigabytes. There are three problems with the default discovery mailbox that is greater than GB:
Not supported.
Unable to migrate to Office 365.
If the default discovery mailbox is in Exchange Server 2010, you cannot upgrade to Exchange Server 2013.
1. Run the following command to delete the default discovery mailbox
Remove-mailbox "discoverysearchmailbox{d919ba05-46a6-415f-80ad-7e09334bb852}"
2. In the message that asks you to confirm whether you want to delete the mailbox and the corresponding Active Directory user object, type Y, and then press Enter.
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When you create a discovery mailbox in the next step, a new user object is created in Active Directory.
3. Run the following command to re-create the default discovery mailbox.
New-mailbox-name "discoverysearchmailbox{d919ba05-46a6-415f-80ad-7e09334bb852}"-Alias "DiscoverySearchMailbox{ d919ba05-46a6-415f-80ad-7e09334bb852} "-displayname" Discovery Search Mailbox "-discovery
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4. Run the following command to assign the Discovery Management role group permissions to open the default discovery mailbox and view search results
Add-mailboxpermission "discoverysearchmailbox{d919ba05-46a6-415f-80ad-7e09334bb852}"-User "Discovery Management"- AccessRights Fullaccess-inheritancetype All
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This article from "Robin's Home" blog, declined reprint!
Delete and re-create the default discovery mailbox in Exchange