Moving the Discovery Search Mailbox in Exchange 2010 is a relatively simple process. Chances are you are trying to remove the default database created by Exchange 2010 setup and it won’t let you because of this mailbox. Finding and moving the arbitration mailbox is outlined below.

Get a list of arbitration mailboxes:

Get-Mailbox -Arbitration

Move the arbitration mailboxes:

Get-Mailbox -Arbitration -Database db1 | New-MoveRequest -TargetDatabase db2

Exchange 2010 - Moving Discovery Search Mailboxes