Gray is actually a legitimate descendant of Arturia, but she was born in a village that dedicated itself to reviving Arturia in the belief that she would lead them into a new utopian society.
To that extent they magically altered Gray's body to make her into a physical match for Arturia. This process was very painful, and although the village was happy about suceeding, Gray hates her new face, so she often covers it with a hood.
It's one of the reasons she likes working for Waver. He likes Gray herself well enough, but he hates her face as it brings up bad memories of the 4th Grail War for him.
