To the best of my knowledge there is no such term. There is no official ceremony nor a name for it. The apprentice (who does not necessarily have to be a Jedi who turned from the light) pledges himself to the Sith, and the Sith Master chooses a Sith name for his apprentice and gives him the title "Darth". This is how Anakin became the Sith Lord Darth Vader:
This is the text of that scene:
Anakin: I pledge myself to your teachings. To the ways of the Sith.
Palpatine: Good. Good. The Force is strong with you. A powerful Sith you will become. Henceforth, you shall be known as Darth . . . Vader.
Episode III: Revenge of the Sith
Anakin simply pledges himself to the Sith and Palpatine gives him his Sith name.
There is no mention of a term like "sithening" or any sort of conversion ceremony in any of the following Wookieepedia articles which might mention it:
Other than the Episode III scene, the closest descriptions of a conversion that I can find are some quotes from a Legends book, which simply says that the Master "chooses" the apprentice. For example:
Bane’s Rule of Two ensured that the power of both Master and apprentice would grow from generation to generation until the Sith were finally able to exterminate the Jedi and usher in a new galactic age. That was why Bane had chosen Zannah as his apprentice: she had the potential to one day surpass even his own abilities.
Dynasty of Evil, p. 9
The Master will train his apprentice in the ways of the Sith, until one day she must challenge him. If she proves unworthy, the Master will destroy her and choose a new apprentice.
Dynasty of Evil, p. 153
In summary, a conversion to the Sith would therefore simply be described as a "pledge", or a Master "choosing" an apprentice, or perhaps an "entitling" since the apprentice gains the title of "Darth".
(Also, the term "Christening" is not limited to Roman Catholicism -- it is used by many Christian denominations.)