And crusade has religious conotations, and the players civ might not be strong on religion... How about 'Terrian Watch?'
I'm thinking more "a remedial enterprise undertaken with zeal and enthusiasm" than "the Crusades". But then the Crusades themselves were remedial in nature despite common misperceptions: They followed centuries of Muslim military aggression and conquest. The religious aspect was a means to stir up that "zeal and enthusiasm". And the DLC is called Crusade after all.
But I do like "Terran Watch". Brings to mind "vigilance". Not bad, but I think "Terran Crusade" better suits the more aggressive, avenging disposition of the faction.
Quoting Rhonin_the_wizard,
The Terran Resistance is probably those that weren't on Terra or entered the pocket dimension. The ones that were led by Jenna Casey and have been fighting against the Drengin.
I get the impression that the Resistance is the new Earth-based faction from this blurb in the announcement:
"In Galactic Civilizations III: Crusade, the shield around Earth has fallen, but humanity has been anything but idle. Furious that their attempts at diplomacy and goodwill were met with treachery and violence, Earth turned to its most valuable asset: its people. From spies trained in espionage to soldiers ready to invade enemy planets, the galaxy is ready for a crusade."
The current Terran Alliance faction will probably remain available for previous campaigns and custom galaxies, but I doubt that there will be two separate Earth-based factions in the Crusade story. Don't know for sure tho, of course.
Jenna Casey looks great btw! Could be me protecting expectations, but she has a "Don't mess with us or you'll regret it!" thing going on. 
