Firstly, mockingjays were seen as a symbol of resistance to the Capitol since the first rebellion: They’re funny birds and something of a slap in the face to the Capitol. During the rebellion, the Capitol bred a series of genetically altered animals as weapons. The common term for them was muttations, or sometimes mutts for short. One was a special bird called a jabberjay that had the ability to memorize and repeat whole human conversations. They were homing birds, exclusively male, that were released into regions where the Capitol’s enemies were known to be hiding. After the birds gathered words, they’d fly back to centers to be recorded. It took people awhile to realize what was going on in the districts, how private conversations were being transmitted. Then, of course, the rebels fed the Capitol endless lies, and the joke was on it. So the centers were shut down and the birds were abandoned to die off in the wild.
Only they didn’t die off. Instead, the jabberjays mated with female mockingbirds creating a whole new species that could replicate both bird whistles and human melodies. They had lost the ability to enunciate words but could still mimic a range of human vocal sounds, from a child’s high-pitched warble to a man’s deep tones. And they could recreate songs. Not just a few notes, but whole songs with multiple verses, if you had the patience to sing them and if they liked your voice.
Secondly—and this is something that was left out of the movie—every tribute is allowed to bring a symbol of their district into the arena with them, as long as it's not a weapon. It's something that allows the audience to remember and identify each tribute, and is supposedly supposed to allow the tribute to better "represent" and show pride in their district. In the books, Katniss' Mockingjay pin is given to her by her friend Madge, who is the daughter of the Mayor of District 12, after Katniss is selected as a tribute, specifically to serve as Katniss' token from her district
So at the beginning of the 74th Games, the mockingjay is already seen as Katniss' emblem—her mascot, so to say—by all of Panem. And it's an emblem that is seen as 'daring' or 'challenging' to the Capitol because the original mockingjays were an embarrassment and a reminder of one of their defeats during the rebellion.
After Katniss wins, some of her "fans" in the Capitol start wearing mockingjay accessories—much as sports fans might wear the mascot of their favorite team. Because the manner in which Katniss won was seen in the districts as an act of defiance, the districts copy this—with a different intention. Essentially, the mockingjay becomes an emblem for the resistance both because of its past history, and because people using it can claim that they're merely supporting a victor of the Games, and thus supporting the capitol. And once Katniss is selected again as a tribute the next year, the fad takes off overwhelmingly, providing even greater cover for those using it as a sign of resistance.
