Hi Boris,
Thanks for asking!
If the MSOLUser still exists (this means that the date on which you moved the MBX to Inactive state is less than 30 days ago), you should restore it, assign an EXO license and the magic will happen.
If the MSOLUser do not exist anymore (soft-deleted more than 30 days ago, or manually hard-deleted), the answer to your question is documented on:
- Recover --- https://docs.microsoft.com/en-us/office365/securitycompliance/recover-an-inactive-mailbox
- Restore --- https://docs.microsoft.com/en-us/office365/securitycompliance/restore-an-inactive-mailbox
My recommendation will be to try first with the recover option (New-Mailbox -InactiveMailbox). In the situation in which customer is in Hybrid, they should do the recover, and after that do a Hard-Match of the MSODS user with the On-Prem user (set the correct ImmutableID on the MSODS user), and just after that sync the user from On-Prem, using AADConnect tool.
If the recover will not work, you should try the restore one (New-MailboxRestoreRequest). In the situation in which customer is in Hybrid, I assume that the steps that they are taking are: create the On-Prem user (enabled as RemoteMailbox), sync it to MSODS (this will create the new MBX in EXO), restore the content of the old MBX to the new one (using New-MailboxRestoreRequest command)
Even so, based on the customer needs, it is up to them which method will use to reaccess data from the old mailbox (recover or restore).