When they had first come to King's Landing, she used to have bad dreams about getting lost in the castle. Father said the Red Keep was smaller than Winterfell, but in her dreams it had been immense, an endless stone maze with walls that seemed to shift and change behind her. She would find herself wandering down gloomy halls past faded tapestries, descending endless circular stairs, darting through courtyards or over bridges, her shouts echoing unanswered. In some of the rooms the red stone walls would seem to drip blood, and nowhere could she find a window. Sometimes she would hear her father's voice, but always from a long way off, and no matter how hard she ran after it, it would grow fainter and fainter, until it faded to nothing and Arya was alone in the dark.
An ominous dream from Arya which is unique in the series so far. The walls of the Red Keep dripping blood certainly makes sense, given what happens. Hearing her father and never being able to come close to him echoes how she felt when she saw him about to be executed. Her wandering, lost and alone, and her inability to find a window is especially interesting. One interpretation of it may be that it shows that Arya does not know where she will be going in a broader sense. Her story following A Game of Thrones certainly supports this, as her journeying has moved her around half the Riverlands and yet in the end she ends up sailing in a direction she never intended before without any knowledge as to the purpose.