This side quest becomes available during the Attend Your First Day At Hogwarts main quest while doing your first assignment for Professor Ronen. Make your way to the Defense Against the Dark Arts Tower in the Astronomy Wing to find Zenobia Noke standing near a statue. Speak with her to learn her Gobstones were stolen and hidden around the grounds. After agreeing to help her, Zenobia will say that her 6 Gobstones can be found on high hard-to-reach areas around the school ground.
Gobstone Locations
The Astronomy Wing – Transfiguration Courtyard
This Gobstone can be found on the roof of a walkway along the eastern side of the courtyard near the Floo Flame. Use the Accio spell to nab it.
The Library Annex – Divination Classroom
This Gobstone can be found in a cannelure hanging in the stairwell across from the ladder leading up to the Divination Classroom. Use Accio on the stone to collect it.
The Library Annex – Divination Classroom
From the Floo Flame, head through the doorway on the right. You can find the Gobstone resting on a rafter. Use Accio to collect it.
The Grand Staircase – Ravenclaw Tower
Travel to Ravenclaw Tower in the Grand Staircase to find this Gobstone on a high ledge in the same hallway as the Floo Flame.
The Grand Staircase – Trophy Room
When you reach the Trophy Room, this Gobtsone can be found high up on a ledge above a window across from the Floo Flame.
The Grand Staircase – Trophy Room
This Gobstone can be found on a shelf along the northern wall in the trophy room.
Rewards
Once you’ve collected them all, return to Zenobia. You have the option of giving her the Gobstones or keeping them. If you hand them over, she won’t learn her lesson and runs off to play with her classmates in the common room again. If you don’t hand them over, she’ll of course be mad at you. Either way you’ll receive the Orbicular-Violet wand handle.