The Name field is used when you create a mail contact via Exchange Online Powershell. (If you create the mail contact in Exchange Admin Center, it will share the same value as DisplayName)
Example cmdlet:
New-MailContact -Name "Chris Ashton" -ExternalEmailAddress "******@tailspintoys.com"
It is used to specific the unique identity of the mail contact.
To modify the Name field, you may need to connect to Exchange Online Powershell as an Exchange admin and use Set-MailUser cmdlet.
Example cmdlet:
Set-MailContact -identity "name of this contact" -Name "unique new name"
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