First, a little refresher: what is a mockingjay?
Well, a mockingjay is a bird, of course, but a hybrid one. They originate from a species known as the jabberjay. Jabberjays were birds that the Capitol genetically engineered to be spies during the rebellion of the past 13 districts. The little guys were meant to overhear subversive conversations, and bring that information to the Capitol, and repeat what they heard. This was all fine and dandy, of course, until the rebels caught on and started feeding the jabberjays false info. Once the Capitol figured this all out, the birds became useless to them. The jabberjays were left alone in the wild where they mated with female mockingbirds. After a few generations, the hybrid birds could no longer repeat or carry information, but they instead learned to carry a tune and to repeat songs and melodies. Hence, the mockingjay was born.
Katniss, like the mockingjay bird, must repeat others' words instead of her own. But she's sure trying to break away from being manipulated and find her own voice:
What they [President Coin and the rebels] want is for me to truly take on the role they designed for me. The symbol of the revolution. The Mockingjay. […] I won't have to do it alone. They have a whole team of people to make me over, dress me, write my speeches, orchestrate my appearances – as if that doesn't sound horribly familiar – and all I have to do is play my part. (1.28)
During the Hunger Games, Katniss became associated with the mockingjay symbol because of the mockingjay pin she was given to take into the Hunger Games arena. This pin became very important to her. The connection between her and the mockingjay idea was strengthened by the final costume Cinna created for her. When Katniss put it on, she embodied the mockingjay. Both Katniss and the mockingjays are mouthpieces that want, ultimately, to find their very own voices. The mockingjays might be genetically unable to do so, but Katniss isn't.
In an interview with School Library Journal, Mockingjay author Suzanne Collins describes the particular role of the mockingjays:
Now the thing about the mockingjays is that they were never meant to be created. They were not a part of the Capitol's design. So here's this creature that the Capitol never meant to exist, and through the will of survival, this creature exists. (source)
In other words, mockingjays aren't supposed to be here, but somehow are. Despite all odds, they're still around. In that sense, it's easy to see why the rebels select the mockingjay as a symbol for themselves: they aren't supposed to be around, yet they persist. And they're determined to survive no matter what.
It's also logical that the rebels would see Katniss as the perfect embodiment of their mockingjay symbol. In fact, Collins goes on to explain that the mockingjay, as an unusual survivor, is a lot like Katniss:
Symbolically, I suppose, Katniss is something like a mockingjay in and of herself. She is a girl who should never have existed. And the reason she does exist is that she comes from District 12, which is sort of the joke of the 12 districts of Panem. Katniss is the mockingjay. She is the thing that should never have been created, that the Capitol never intended to happen […]. (source)