Normal Office 365 accounts would only have a 50gb mailbox, any more then this and I would recommend enabling Archiving which can go to 1TB

1. Add Trial verison of a license with Exchange Online Plan 2 which increases their Mailbox size to 100GB ( e.g.M365 E5 ) for a month for Migration. Assign this to the user to get their Mailbox size to 100GB
2. Migrate the email with Bit-Titan can do a prestage and only do items old then a couuple of years to start with,
3. Wait for the Sync, if you synced over 100gb it will come up with a failure 
4. Enable Archive on the 365 Mailbox. Archiving runs every day , you can force start with this
5. If the source mailbox is below 100GB you can enable archiving to drop it back down to 50gb then remove the license.
6. If the source mailbox is over 100GB , let the Archiving finish , then rerun the Sync until its completed ( It will make a note in the database of items its already replicated )