What were the names of the governors of Sedom, and what is the significance of those names?
Or perhaps you mean the kings of Sodom and its sister cities?
Their names were: Bera, Birsha, Shin'av, Shem'ever, and one whose name is not recorded (Ramban says that this is because he wasn't famous, as he ruled over the small town of Tzoar).
The significance of their names, according to Rashi:
- Bera - ב' רע, doubly bad - against Hashem and against humanity
- Birsha - נתעלה ברשע, superbly wicked
- Shin'av - שונא אב, hates his Father in Heaven
- Shem'ever - שם אבר, he made himself "wings" to fly up and attack Hashem
and according to the Midrash (Bereishis Rabbah 42:5):
- Bera - בן רע, an evil son
- Birsha - בן רשע, a wicked son
- Shin'av - שואב ממון, sucking up money
- Shem'ever - שם אבר, flying around to get money
Perhaps you mean this Gemara (Sanhedrin 109b)?
ארבע דייני היו בסדום שקראי ושקרוראי זייפי ומצלי דינא
There were four judges in Sdom: Mr. Liar, Mr.Liarar, Mr. Forger, and Mr. Justice-Perverter.
I think the Gemara's indicating they weren't too honest!