Someone found this blog today by searching for:
song when jaimes hand is cut off
The song you’re thinking of is “The Bear and the Maiden Fair.”
And I’m afraid that song was not included in any of my previous blog posts, but anyway. In the books, the song doesn’t play into the maiming, but on the show, it does. The Bolton soldiers who take Jaime and Brienne captive sing the song on the road during S3:E3, and then the song is performed again during the closing credits, right after the hand-chop. Here is how the song relates to the series. Complete with lyrics! You’re welcome.