Doing a migration for a client last week from Exchange 2003 to Exchange 2010 and when I went to move users mailboxes from Exchange 2003 to Exchange 2010, I got the message below “Mailbox Database… is offline”. It put me in a bit of a spin to start with but I eventually found it was referring to the Exchange 2003 server.
The solution was to add the Exchange 2010 servers computer account to give it Full control access on the mailbox store on the Exchange 2003 server. You can do this on the Exchange 2003 System Manager > First Storage Group > Mailbox Store > Properties > Security Tab. Restart the Exchange 2003 Information Store service afterwards to ensure the change takes.