This is a small domed chamber abutting the southern facade of the Altun-Aba (Iplikci) Mosque, which is thought to belong to the now demolished madrasa of Altun-Aba. Mevlana is said to have stayed here as a teacher at the madrasa. This chamber is all that remains of the madrasa where Mevlana spent several years as a guest with his family during his father's lifetime.
Above text and pictures are from the book titled "Mevlana and the Mevlana Museum".
You can purchase "Mevlana and the Mevlana Museum" book and other Turkey related books from Explore Turkey Bookstore.