Once a ninth world is discovered orbiting the Sun, the discoverer suggests a name to the heads of two groups within the IAU: the Working Group for Planetary System Nomenclature and the Working Group for Small Bodies Nomenclature. They must agree on a single name, (which let's face it, will probably be a Roman or Greek god.)
The heads of the two working groups return to their respective committees and further discuss and debate. Among the things they consider is how does a name sound in languages other English? Is it offensive to a people or culture? Does it comply with naming conventions? If and when both working groups approve and agree on a name, it moves forward.
Here's where it gets complicated, because no one knows what happens next. The proposed name might then go to the president of the IAU for consideration. It might simply go to a vote of the full body of the IAU. Or, maybe, the IAU will rely on its working groups—but probably not, because who wouldn't want to help name a planet? But once the majority agrees and no one finds fault with a name, only then does it get assigned to Planet Nine.