Let's try to understand the term Azzi. This term was written not only in conjunction with Hayasa but also separately.
It has been suggested that double zz in Hittite was pronounced as ts or tz. So some believe that it was pronounced as Atsi. The first thing that comes in mind is the word atsu (ածու) used by Khorenatsi to denote the term people. This word is also a farming term. It is a native Armenian word.
Looks good except that there is no much evidence of later use of this term.
Another possibility is that the root was the Az-zi while the final zi was a suffix that appear in numerous toponymes.
With the root Az we have some links. Like the Az-ord in Tayk', relatively close geographicaly to Azzi.
In Ptolemy there is an Azata close to both of this terms.
In Urartian texts we have the land Aza in modern Armenia. Close to modern Armavir. And the same Aza is attested in Greek period as Azara. Azara is probably the plural of Aza.
We have also some personal names. Aza king of Mannai. Azo legendary king of Iberia. Also tentatively Asia king of Diau.
That's all. I don't know any other example
Little bit more speculative is the biblical term Uz. Which usually is placed in Syria.
If Uz and that Az are related then this would increase the chance that we are dealing with some kind Mitanni Aryan term. Here we can remember that the Armenian term az/azn (tribe, genus, nation) is usually believed to be a loanword from Indo-Iranian.
If it is a chance similarity then we can forget it.