The Names of Sauron

[Inziladun]

Nice thread, Legate. This is something I've wondered about from time to time.

As you note, Sauron was his name among all those in the West of Middle-earth. He would have had different names in the tongues of the Easterlings and Haradrim, and probably also in the Black Speech, but Sauron he was to all his enemies in the Third Age.
In the Elder Days, all the Valar and Maia known to the Elves of Valinor were given names. Sauron would have been a result of his association with Morgoth.

Of old there was Sauron the Maia, whom the Sindar of Beleriand named Gorthaur. In the beginning of Arda, Melkor seduced him to his allegiance.

Silmarillion Of the Rings of Power of the Third Age

So he'd had the moniker Sauron almost from the first. If he had been given any other name by the Eldar before that, it would only have been used a relatively brief time, before his service to Melkor became known. His 'original' name would possibly have made a reference to Sauron's beauty, or perhaps his service to Aulë (and I'm admittedly just guessing here). But would a name making reference to his being good have been appealing to him in the Third Age, when he was well aware he could no longer fool the West by appearing to be beautiful and good? On the contrary: he reveled in evil, relished the thought of crushing the West. Therefore, I think being called the abhorred was something that could have given him some perverse pleasure.
As to the Mouth's name in Sauron's forces, I think it might actually have been 'Mouth', or 'the Speaker', or something like that. After all, his original identity had been swallowed up, so what else would they have called him?