So, there was not really a "simple" solution. The exact details stay unresolved.
What we did, was setting up a new Exchange 2010 Server on a new machine (and, for good measure, an additional domain controller on yet another new machine), and then move the mailboxes from the Small Business Server to the Exchange 2010.
When we got to the problematic mailbox, it refused to move with the following message:
07.01.2013 19:01:07 [exchange-2010] Initializing folder hierarchy in mailbox 'Primary (76582dc1-1a82-428e-bd05-e1276800e425)': 105 folders total.
07.01.2013 19:01:17 [exchange-2010] Fatal error MapiExceptionPartialCompletion has occurred.
Error details: MapiExceptionPartialCompletion: Unable to copy to target. (hr=0x40680, ec=0)
at Microsoft.Mapi.MapiExceptionHelper.ThrowIfErrorOrWarning(String message, Int32 hresult, Boolean allowWarnings, SafeExInterfaceHandle iUnknown, Exception innerException)
at Microsoft.Mapi.MapiUnk.ThrowIfErrorOrWarning(String message, Int32 hr)
at Microsoft.Mapi.MapiProp.CopyTo(MapiProp destProp, Boolean reportProgress, CopyPropertiesFlags copyPropertiesFlags, Boolean copySubObjects, ICollection`1 excludeTags)
at Microsoft.Mapi.MapiProp.ExportObjectHelper(IMapiFxProxy dest, PropTag[] tags, CopyPropertiesFlags copyPropertiesFlags, Boolean useCopyProps)
at Microsoft.Mapi.MapiProp.ExportObject(IMapiFxProxy fxProxy, CopyPropertiesFlags copyPropertiesFlags, PropTag[] excludeTags)
at Microsoft.Exchange.MailboxReplicationService.LocalSourceFolder.Microsoft.Exchange.MailboxReplicationService.ISourceFolder.CopyTo(IFxProxy destFolderProxy, PropTag[] excludeTags)
at Microsoft.Exchange.MailboxReplicationService.SourceFolderWrapper.<>c__DisplayClass1.<Microsoft.Exchange.MailboxReplicationService.ISourceFolder.CopyTo>b__0()
at Microsoft.Exchange.MailboxReplicationService.ExecutionContext.Execute(GenericCallDelegate operation)
at Microsoft.Exchange.MailboxReplicationService.SourceFolderWrapper.Microsoft.Exchange.MailboxReplicationService.ISourceFolder.CopyTo(IFxProxy destFolder, PropTag[] excludeTags)
at Microsoft.Exchange.MailboxReplicationService.MailboxCopierBase.CopyFolderProperties(FolderRecWrapper folderRec, ISourceFolder sourceFolder, IDestinationFolder destFolder, FolderRecDataFlags dataToCopy)
at Microsoft.Exchange.MailboxReplicationService.MoveBaseJob.<>c__DisplayClass36.<>c__DisplayClass38.<CreateFolderHierarchy>b__31()
at Microsoft.Exchange.MailboxReplicationService.ExecutionContext.Execute(GenericCallDelegate operation)
at Microsoft.Exchange.MailboxReplicationService.MoveBaseJob.<>c__DisplayClass36.<CreateFolderHierarchy>b__30(FolderRecWrapper folderRec, EnumFolderContext context)
at Microsoft.Exchange.MailboxReplicationService.FolderMap.EnumSingleFolder(FolderRecWrapper folderRec, EnumFolderContext ctx, EnumFolderCallback callback, EnumHierarchyFlags flags)
at Microsoft.Exchange.MailboxReplicationService.FolderMap.EnumSingleFolder(FolderRecWrapper folderRec, EnumFolderContext ctx, EnumFolderCallback callback, EnumHierarchyFlags flags)
at Microsoft.Exchange.MailboxReplicationService.FolderMap.EnumSingleFolder(FolderRecWrapper folderRec, EnumFolderContext ctx, EnumFolderCallback callback, EnumHierarchyFlags flags)
at Microsoft.Exchange.MailboxReplicationService.MoveBaseJob.<CreateFolderHierarchy>b__2f(MailboxMover mbxCtx)
at Microsoft.Exchange.MailboxReplicationService.MoveBaseJob.ForeachMailboxContext(MailboxMoverDelegate del)
at Microsoft.Exchange.MailboxReplicationService.MoveBaseJob.CreateFolderHierarchy(Object[] wiParams)
at Microsoft.Exchange.MailboxReplicationService.CommonUtils.CatchKnownExceptions(GenericCallDelegate del, FailureDelegate failureDelegate)
Error context: --------
Operation: ISourceFolder.CopyTo
OperationSide: Source
Primary (76582dc1-1a82-428e-bd05-e1276800e425)
PropTags: [ContainerContents; FolderAssociatedContents; ContainerHierarchy; RulesTable; 1071710466]
--------
Folder: '/Top of Information Store/Inbox', entryId [len=46, data=00000000B55449F4D13FC54AAA4A5936050753AE01005896698ABF45814BA7D90A6A0EA2FEAF00000000329A0000], parentId [len=46, data=00000000B55449F4D13FC54AAA4A5936050753AE01005896698ABF45814BA7D90A6A0EA2FEAF0000000032970000]
07.01.2013 19:01:18 [exchange-2010] Removing target mailbox 'Primary (76582dc1-1a82-428e-bd05-e1276800e425)' due to an offline move failure.
07.01.2013 19:01:18 [exchange-2010] Relinquishing job.
We took this as the final evidence that there is really something very wrong with that mailbox. So we deleted it on the old server, created a new one, and are now copying back all the needed mails, calendar entries, contacts, notes and todo items from Outlook.