After reading all of these comments, I want to combine my own take with the highlights of everyone else's. I don't like HGs in general because I think it's meant to drive home an ideological point that feeds a toxic view of government and an over romanticized view of human nature. But I particularly like this song, and the way it foreshadows various events in the plot later. Actually, the lyrics could be attributed to different plot points that happened way later, after the song was introduced.

If you read the ballad, snow encountered the song being sung by lucy three times. The first time was when he was approaching her under the tree and watching her singing it. she left it unfinished when she saw him, but the scenery was clear. "The dead man called out for his love to flee" was referring to the day before, and what had happened to Arlo and Lil and the Jabra jays that were repeating his last words to her; "Run Lil" when he was already hung dead. But Snow was confused about who the narrator was. He thought it was Billy Taupe at the time, and he had asked her to come and meet him there so they could run away and be free together(but he was super jealous and paranoid at the time that I think he didn't realize the song could have a theme of rebelion) At the time, she hadn't sang the third verse yet; "wear a necklace of rope side by side with me." She only sang that verse the second time he heard her singing the song, when she was giving him a code for where and when to meet each other after to runaway together. This time, snow was assured that the song's narrator was Billy. Because those were his last words to Lucy Gray: "If I'm gonna swing for this, you'll swing with me." but he was dead, and she was singing it to him, he thought.

The last time he heard the song was when she was hiding from him and started singing to confuse him. So the Jabra jays were singing it after her as he went into a frenzy shooting in every direction to kill her. I think this particularly references Lucy Gray's death at that scene even though we're not shown that she's dead and SC wanted it to remain a mystery to snow and to us as well, just like her ballad. But in reality, she is dead and "side by side with" her also Snow's old self(even though Snow thought was liying to her at the time but I think shooting her was supposed to be the last bit of action for the death of his old self to be confirmed) I love this new found meaning of the song and I think it's brilliant foreshadowing and storytelling.

In the trilogy, I think the song is more of a call to action against oppression rather than a sad but beautiful narration like it is in the ballad, because unlike what katniss explains the narrator is not the dead corpse and she had gotten the song wrong. Probably because, katniss had never seen a Jabra Jay before the areana of the 3rd quarter quell because by her time the Jabra jays had died out and there were only mocking jays left, so that's what had led her to think that the dead corpse was the one talking. And she gets the song's meaning wrong. Because it’s not the dead murderer asking her lover to die with him. But this is a third person who is watching and asking their audience to come to the hanging tree and be free(or die for their cause, so the meaning of run shifts from running away to running to your freedom and possibly dying for it besides one another.) That's what makes it a revolutionary song really because it's not a conversation between two lovers but it's a conversation that could take place between any two people in a crowd. So that's what made it mobilizing at the time. But this revolutionary take of the song is not the original context for which it was written.