The name Cisco Unity Connection Messaging System is actually the display name for the user with the alias UnityConnection. Unfortunately, it does not appear to be editable via the GUI.
If you are brave enough, it can be modified via CLI. I do not know if it is officially supported, but it is technically.
From the CLI you can execute the following to see the user in question:
run cuc dbquery unitydirdb select displayname,objectid from tbl_user where displayname = 'Cisco Unity Connection Messaging System'
You should be returned something like:
displayname objectid
--------------------------------------- ------------------------------------ C
Cisco Unity Connection Messaging System aeb80eab-1ec0-4add-8d4f-f29396639324
Then, cross your fingers and execute the following. Here for example I am changing the name to The ABC Company:
run cuc dbquery unitydirdb update tbl_user set displayname = 'The ABC Company' where displayname = 'Cisco Unity Connection Messaging System'
In the GUI you will find the UnityConnection user's display name is now changed to the company name, here for example, The ABC Company. Messages will still show From:Cisco Unity Connection Messaging System though.
I don't know what service(s) restart commits the change, but I had the luxury of rebooting the entire server. After that, the From: address in UM does show the new name you defined.