Updated June 2011: Maisie Williams is a total newcomer who is playing Arya Stark to perfection! She is one of the well casted characters in the show and after seeing her as Arya, I can't imagine anybody else playing this role.
I didn't really know any young female actresses back then when I was trying to cast actors for these characters in my head - the most that I could think of was that girl from the recent version of Charlie and the Chocolate Factory, Julia Winter.
Arya Stark is a spirited young girl with an interest in fighting and exploring. Throughout her travels, Arya displays great resourcefulness, cunning, and an unflinching ability to accept hard necessity. She is skinny, athletic, and often mistaken for a boy. She has strong Stark family features, with a long face and brown hair. She is said to take after her fiery aunt Lyanna in temperament and appearance. Her direwolf companion is named Nymeria.
For Arya, I actually pictured a young Natalie Portman from the movie Leon or The Professional. I suppose if they were to create the TV show, maybe that spoilt rich girl - Julia Winter, from Charlie the Chocolate Factory might be alright.