Girl-on-a-Dragon (originally announced as Realder's Dragon) was a stone-dragon named after the Skill-coterie member who gave heart to it, Realder. It stayed incompleted because Salt, the leader of the coterie carving it, didn't want to lose her individuality and tried to keep herself separated in the girl part of the statue. Nighteyes' name for the creature was Cold One.
When seeing her in the stone quarry, the Fool felt sorry for her and started carving out the Stone-dragon with his Skill-touched fingers. Fitz helped him by putting some of his memories in the stone. After Verity-as-Dragon had awoken and flown away, the Fool was back at the statue when Burl attacked and wounded him. Nighteyes then attacked Burl, who died bleeding all over the stone, thus awakening it. Awakened, her scales are green and shine like dark emeralds in sunlight.
Girl-on-a-Dragon then took to wing with the Fool on her back. They briefly returned to aid Fitz who was battling Will and Regal's men before flying away with the rest of the awakened Stone-dragons.
Note: At the end of Assassin's Quest, Fitz and the Fool both think one of the completed stone dragons in the garden was Realder's dragon. By the end of Fool's Fate, though, it has become clear that Girl-on-a-Dragon is the one Realder gave his life to.