A dump of the Japanese Version of the Dark Souls 2 Network Test reveals unused dialog that was never translated into English. I was recently made aware of this by an anonymous tipster, though it turns out it’s been known and linked on ebiflynageruyo’s blog this whole time!
Judging from the dialog it seems like this quest was likely cut due it requiring some sort of escort mechanic. In the final game all NPCs are either static points on the map or silent phantoms summoned to help with specific bosses. This dialog suggests the player would be escorting the young girl through some area, which would introduce the difficulty of pathfinding for her (though they could simply use the same pathfinding that the phantoms use). Also, since souls games allow you a great degree of freedom, they would have to address the possibility of the player attacking and killing the young woman; violence against children is flatly not allowed in games released in America, so that could have been a sticking point in deciding to cut that content.