Vault Dweller Good news, The New World was a terribly dry title.
Too bad that your 3 options are even more terrible.
The obvious thing is to name the game after the ship. But 'Starfarer' is just a godawful name for a starship, especially one that's supposed to be the scene of exciting adventures. Don't they have PR people in the future? In terms of uninspired fantasy phraseology, this is close to hŵrpa dŵrp territory. You are better than this.
So what I suggest is rename the ship to something more compelling, and then name the game after the ship. The ship could be named after a great explorer, some other great figure with a slightly ominous undertone, as was mentioned, after a star. So, for example:
Starship Arcturus
Starship Vega/Chara/Betelgeuse
Starship Aldebaran (or just Aldebaran)
Starship Magellan
Even better if you can subtly allude to the quasi-postapocalyptic situation through the ship's name. Like the ship in Alpha Centauri was called Unity. Maybe something playing on your religious themes? Maybe your ship could be called Hope, Faith, Deliverance (hey, that's not so bad, if it wasn't already taken - how about 'Starship Deliverance'???)... nah, the title should probably not have 'Starship' in it, to make it more impactful. I think I'm onto something though, let's look through the thesaurus...
Salvation?
Redemption?
Savior?
Anything but 'Starfarer' please. That is such a dull and meaningless name that it actively makes me lose faith in your project, lol. Why would I trust a game author that invented such a shitty, artless name to tell an interesting story? At least make an effort.