The girl is Ned's sister Lyanna Stark, ( so in that scene you are seeing Ned, older brother, fighting Brandon, Ned's younger brother, and Lyanna rides in on the white horse, Hodor is the young boy that is a big kid, Bran is up with the old man basically seeing the past before Bran is even born) in the books before the HBO series Lyanna is kidnaped by the mad king's son Prince Rhaegar Targaryen, where it is written that he rapes and hypothicized he impregnates her. When the Mad King is killed by Jamie Lanister(king slayer) Ned finds her in a "Bed of blood" and she makes him swear to a promise which all that is written "Promise me, Ned". It has also been written that male Targaryen's that are born always kill the mother. They can't take the trauma of a male Targaryen birth. Many like myself, don't believe Mr Morales Ned Stark ever had an affair, that the promise is to keep secret the true lineage of Lyanna Starks child, we think to be Jon Snow, which makes him a Stark and Targaryen. Also Tyrion killed his mother upon birth, and can now be close to Dragons, so Joanna Lanister (Cercei and Jamie's mother) must of had an affair or raped by a Targaryen. Making Tyrion Lannister a Targaryen also.
Lyanna Stark
http://m.westeros.org/index.php/Lyanna_Stark
FYI Eddard Stark is Ned Stark.