The cited answer is correct insofar as the IAU is not the body that makes names of celestial bodies official. However, it's authority is implicit because it is recognized by the astronomical community and the public as the foremost authority on astronomical naming, and thus the names the IAU adopts are the ones most commonly used in the scientific community (excluding objects referred to by their catalog numbers).
From one of their information pdfs:
The IAU has been the official arbiter of planetary and satellite naming since its inception in 1919. The IAU’s decisions are officially adopted by the nearly 11 000 professional astronomers who are its members, coming from more than 90 countries.
. . .
The IAU does not consider itself as having a monopoly on the naming of celestial objects — anyone can in theory adopt names the way they choose. However, given the publicity and emotional investment associated with these discoveries, worldwide recognition is important and the IAU offers its unique experience for the benefit of a successful public naming process (which must remain distinct, as in the past, from the scientific designation issues).
The IAU may reach out to the public for naming suggestions, or it may taken case-by-case suggestions by individuals. So while Batygin and Brown can call the ninth planet whatever they want - they've informally referred to it as "Phattie" - public consensus will largely rest on the results of the IAU's decision, which could, of course, end up taking a suggestion of the pair.
Anyone can come up with a suggestion, although sometimes that's a terrible idea.