Active Directory: Handle Suffixes in a User's Last Name?

Solution 1:

The Active Directory Schema defines an attribute Generation-Qualifier for exactly this purpose.

Your employer may decide to include the suffix string in other attributes such as Canonical-Name, Sam-Account-Name, User-Principal-Name. You may even make modifications to the Configuration partition to "automate" such inclusions, but that is outside the scope of your question. See Default full name format in Active Directory for an example.

Solution 2:

It's your environment, so I would do whatever makes you feel comfortable and would make it easier to remember. For me personally, I would add jr and use number 3 instead of III.