It was supposed to have been The Queen's House, built in 1530.
Anne stayed in the royal palace complex (now demolished) situated between the White Tower and the Lanthorn Tower.
In her jailer William Kingson's report, he said that Anne stayed in the same lodgings as she did during her coronation (that is the palace area).
There is an interesting video about exactly this on youtube. The link is http://www.youtube.com/watch?v=HZcYI2D2a9s&feature=related.
Yes, she stayed where the South Lawn now is, beside the White Tower. The Queen's House wasn't actually built until the early 1540's.Jane Grey stayed there but not Anne.
Just as a useless sidenote, contrary to myth, that is only named that because there is a current queen on the throne. If Prince Charles became monarch, it would become the King's House, because technically that is where he would stay if he ever fancied it. USeless fact for you ;) ).