Last week out of curiosity I used the permanent invisibility glitch on Regret on Legendary so I could examine Regret's throne. The key thing I noticed is the yellow bands of plasma that the throne emits. These bands of plasma are of the same type that 343 Guilty Spark emits. I believe the thrones' floating technology was reverse-engineered from that of a Monitor.
At the bottom of the throne is a metal block which appears to be a counter-weight, meant to shift the center of gravity upward so the Prophet does not spin like a gyroscope when something hits his throne (LOL), but only a slight bobbing which stops eventually.
This bobbing can be seen in the Arbiter cutscene, when the Prophet of Mercy slams his fist on the right-side arm rest ("Were it not for the Arbiters, the Covenant would've broken long ago!"), his throne bobs slightly then stops.