Who wrote the lyrics and the music to "The Rains of Castamere" aka the Lannisters song

Question

The song The Rains of Castamere is performed by Bronn and the Lannisters soldiers (S2E9). It's also played by the band in The red wedding (S3E9).

After finishing singing a soldier of the Lannister's asks Bronn where did he learn a Lannister's song?

Come to think of it that's a very good question since in the books there's no melody (tune) to this song.

GoT - Bronn and Lannisters soldiers singing "The Rains of Castamere" (S2E9)

Answer 1

TL;DR: Out of universe, the lyrics are from the books, and the music is by Ramin Djawadi.

This is part of my answer to a related question on another Stack Exchange; this was based on this article detailing "the origin story behind the most iconic song on Game of Thrones":

"Rains of Castamere" is the most well-known piece of music in George R.R. Martin's "A Song of Ice and Fire" universe. The song tells the story of how House Lannister destroyed the rival House Reyne of Castamere when they dared to rebel. In the years that followed, "Rains of Castamere" would play to both celebrate House Lannister and strike fear in the hearts of their enemies. The tune was famously played at the beginning of the Red Wedding, signaling the Freys alliance with the Lannisters and cuing the massacre's start.

Given the rich history of the song, its onscreen debut in "Game of Thrones" was a widely anticipated moment for fans of the book series. But there was a catch: No one knew what the actual melody of the song would sound like.

[...]

"Benioff and Weiss said there's lyrics in the books for 'The Rains of Castamere' that we would like you to write a melody for," Djawadi tells INSIDER. "It needed to be haunting but beautiful, and all these other criteria because the theme had to be really flexible. So just based on the lyrics I sat down and wrote it."

Answer 2

A Song of Ice and Fire

No specific mention is given to who actually wrote the song other than it being made by "singers".

This Westerling betrayal did not seem to have enraged his father as much as Tyrion would have expected. Lord Tywin did not suffer disloyalty in his vassals. He had extinguished the proud Reynes of Castamere and the ancient Tarbecks of Tarbeck Hall root and branch when he was still half a boy. The singers had even made a rather gloomy song of it. Some years later, when Lord Farman of Faircastle grew truculent, Lord Tywin sent an envoy bearing a lute instead of a letter. But once he'd heard "The Rains of Castamere" echoing through his hall, Lord Farman gave no further trouble. And if the song were not enough, the shattered castles of the Reynes and Tarbecks still stood as mute testimony to the fate that awaited those who chose to scorn the power of Casterly Rock. "The Crag is not so far from Tarbeck Hall and Castamere," Tyrion pointed out. "You'd think the Westerlings might have ridden past and seen the lesson there."
_{A Storm of Swords, Tyrion III}

However, we do know that it must be a very recognisable tune as Catelyn is able to identify it without the words.

With scarcely a moment's respite, they began to play a very different sort of song. No one sang the words, but Catelyn knew "The Rains of Castamere" when she heard it.
_{A Storm of Swords, Catelyn VII}