The Hand's study.
You've been holding the council meetings in the Tower of the Hand." ―King Joffrey Baratheon to Lord Tywin Lannister
Tower of the Hand is a location within the Red Keep in King's Landing. It serves as the residence of the Hand of the King and his household. Among other rooms, it contains a dining room, the Hand's study, the Hand's bed chamber and connecting corridors and stairways.
During his short tenure as
Hand of the King, Eddard Stark takes residence in the tower alongside his daughters Sansa and Arya.
Tyrion Lannister takes up residence in the tower when he assumes the role of Hand of the King. Secretly, he also installs his lover Shae with him, until she's discovered by  Varys.
Tyrion also has dinner with
Janos Slynt in the Tower before exiling him to the Night's Watch. Cersei visits Tyrion in his bed chamber to confront him about his treatment of Slynt.
After assuming the mantle of Hand of the King,
Tywin Lannister takes residence in the Tower. He also has small council meetings take place in the Tower instead of the small council chamber. 
Behind the scenes
In the audio commentary track for
, The North Remembers D.B. Weiss revealed that falcons representing House Arryn on the wall still remain since neither Ned Stark or Tyrion Lannister have had time to redecorate after Jon Arryn's death.