I'm not sure of the cause, but perhaps the particular groups used to be security groups with Active Directory or something. But thanks to the Microsoft KB, a solution was found.
Essentially I needed to login to the Exchange Online server via Remote Powershell, and then I could use the Set-DistributionGroup cmdlet fix things up. This cmdlet allowed me to assign ownership to the problem distribution group.
These are the references I used to get to this result:
- http://support.microsoft.com/kb/2731947 (The overall solution, explaining the cmdlet options)
- http://technet.microsoft.com/en-us/library/jj984289(v=exchg.150).aspx (how to Connect to Exchange Online Using Remote PowerShell)