In total, Katniss sang 3 songs that had huge significance in her life. These simple songs attracted the mockingjays and like always, her lovely voice was appreciated by recomposing the song in their brilliant manner.
The first song was the valley song. We didn’t know the exact lyrics of the song because she hardly remember the occasion but that particular moment cemented Peeta’s undying feelings for the girl with the red plaid dress.
“And I swear, every bird outside the windows fell silent.” [The Hunger Games, Chapter 22]
The second song was the mountain air song, the lullaby she used to sing for Prim, the same one she sang to Rue during her final moments. Not only she managed to fulfill her last wish, she realized what game the Capitol put them through and sent the message nationwide.
“Then, almost eerily, the mockingjays take up my song.” [The Hunger Games, Chapter 18]
The third song was The Hanging Tree. Caroline McCormick sang it beautifully on the audio book and I found some nice melodic version of it on Youtube too. I particularly liked the way Ms. Lost-on-cloud-9 dissected it here. It was perfect!
“Peeta’s right. They do fall silent when I sing. Just like they did for my father.” [Mockingjay, Chapter 9]
This is also the song in which Peeta showed some significant progress post-hijacking.
Just like the pearl from which we could relate beauty (that arose from pain) and purity with Peeta, the pin connected freedom and rebellion with Katniss. Note that both are gifts given to her and both are lost after Katniss went back to District 12. You can check her box. I tell you the pin and the pearl was gone, probably destroyed in the same fire that scarred Katniss and Peeta for life.
In my opinion, the writer omitted the reappearance of the gifts for some obvious reasons. The pearl symbolized Peeta and he was back in her life. She no longer had to kiss the pearl for missing him so badly. He is her pearl so she won’t be needing a substitute anymore. On the other hand, the pin symbolized the rebellion and there is no need for one at the time being. Panem is free.
I do believe that you never lost anything. It’ll just come back to you dressed in a different suite, different shape, different colors, so many different ways.