From A Wiki of Ice and Fire
Jenny's song is a sad song that is always requested by the ghost of High Heart as payment in exchange for telling the brotherhood without banners of her prophetic dreams. It may be about Jenny of Oldstones, who, according to the tales, was acquainted with a woods witch.
A Storm of Swords
Arya Stark hears the song when the brotherhood takes her to High Heart for the first time. The men meet with the ghost of High Heart, who requests her song after telling them of her recent dreams. Tom of Sevenstreams sings the song, accompanying the words with his woodharp. The song is so soft and sad that Arya can only hear snatches of the words, though the tune is half familiar to her. "Sansa would know it I bet", she thinks.
When the brotherhood go to High Heart a second time, Tom again sings the song for the dwarf woman. As he is singing she closes her eyes and rocks slowly back and forth, murmuring the words and crying. Thoros of Myr takes Arya firmly by the hand and draws her aside, telling her,
|“||Let her savour her song in peace. It is all she has left.||”|
Later, while the brotherhood waits for Merrett Frey to arrive at Oldstones with his brother’s ransom, Tom may have been singing some lyrics of the song while sitting on the sepulchre of King Tristifer IV Mudd. Merrett finds the music is soft and sad, and he thinks of the song's lyrics,
|“||High in the halls of the kings who are gone, Jenny would dance with her ghosts ...||”|
It may be the same song that Robb Stark mentions to his mother Catelyn Tully when he is standing before the sepulchre of Tristifer at Oldstones. When Catelyn tells him the name of the ancient castle where they are encamped, Robb says to her,
|“||There is a song. Jenny of Oldstones, with the flowers in her hair.||”|
|“||She always makes me sing the same bloody song.||”|
|“||Oh, aye. My Jenny’s song. Is there any other?||”|