Hope is something that the mockingjay symbolizes. The mockingjay, a freakish hybrid created from the genetic experiments of the capitol, is something that is beautiful and hopeful that springs unexpectedly out of a tyrannical rule. Just like the mockingjay, Katniss and Peeta offer that hope; they are something strong and good that comes out of the awful measures that the capitol takes against the people. In that sense, the mockingjay also represents Katniss, and the hope and strength that she gives the people. Against all odds, she survives, and creates a fresh new hope for everyone.
It also symbolizes how Katniss and Peeta use the "experiments" of the games against the capitol. The capitol set up an elaborate hunger games tournament, altering the environment and influencing the players, just like they did in creating a new species of bird. But then, enter the unknown factor, the wild creature (the mockingbird or Katniss and Peeta) and it created a hybrid that was able to oust the creators of their own genius. Peeta and Katniss take what the capitol has given them, the games, and infuses them into their own lives, making them their own, just as the mockingjay has done.
I hope that helped; good luck!
The Mockingjay was created when the mutant birds were set free. These mutants were used during the rebellion to spy on the citizens of the district. They were sent out to listen then would fly back to the Capitol to repeat everything they heard. After the rebellion had ended the birds were set free. They were expected to die out. Instead it mated with the mockingbird. This new bird was able to listen and repeat a melody that was sung to it. Other Mockingjays would join in and could carry on for hours. The creation of this new species became a sign of hope, that anything was possible. A sign that through all the cruelty of the Capitol, anything was possible. Before leaving for the games Katniss was given a pin with a Mockingjay on it. In the arena Katniss is faced with many difficult situations, but she is able to stay true to who she is, she never loses that. The Mockingjay becomes a symbol of Katniss, which in that later books, becomes so much more.
In the entire Hunger Games series, symbolism is a very important aspect. There are many different objects, people, and otherwise that represent a multitude of things, including rebellion, love, and strength.
Of course, one the most important and well-known symbols of the entire series is the Mockingjay. Whether it be the pin or the actual bird, they both symbolize the same thing: hope.
The hope that the kids will survive. The hope that the outer districts won't starve to death. The hope that the rebellion will overthrow the Capitol and things will change for the better.
The mocking jay in the Hunger games symbolize Hope, because when district 13 was destroyed, they continue the fight the capitol in the background with the mocking jay as a symbol because it was made by the mocking bird and the blue jay one of which were once used by the capitol but then destroyed.
The mockingjay was a mistake. The jabberjay (during the Dark Ages) were made by the Capitol to mimic the rebels in the districts in order to extract information. However, the rebels noticed and made the jabber jays say whatever THEY wanted them to say. The jabber jays escaped and mated with the mockingbird, thus making the mockingjay. The mockingjay resembles rebellion and something that was supposed to favor the Capitol but didn't. However there is beauty in the mockingjay, because its a bird and it doesn't bother anyone. Mocking jays show that the Capitol can't control EVERYTHING and the things that they think they have control over, they actually don't. The people view Katniss as their mockingjay because she took something so valuable and that resembled the Capitol's power (the Games) and turned it around in a way that showed the Capitol's flaws in power; she made them look weak. The mockingjay is the embodiment of rebellion.
While agreeing wholeheartedly with all the above answers, I feel there is another aspect to the Mockingjay so far only briefly alluded to: communication.
One outstanding quality of the mockingjay are its beautiful and elaborate songs - and the fact that these mostly originate from other sources, which the birds pick up, transmit, and sometimes alter. Mockingjays cannot be abused for spying like their forefathers, the jabberjays, but they can still carry on a melody across wide distances, like an African drum. This in itself can serve as a means of communication: District 11 used it for warnings, Katniss and Rue to reassure one another; and the new Rebellion uses the Mockingjay, i.e. Katniss, as the unifying face and voice in the fight against the Capitol.
As the Mockingjay of the rebels, Katniss is turned into something like a leader-figure; but in the three books, she rarely really acts as a leader. She was not the one to call people to fight in the first place, but her individual (if defiant) actions in the arena of the 74th Hunger Games were amplified into a call for rebellion by the people in the districts (and President Snow himself). Even when she officially accepts the role of the Mockingjay, she is not the one who plans and designs the revolution but their broadcast face and mascot.
In reading the books, the degree to which Katniss's effect in the media is instrumentalised by all sides sometimes stung me. Like the real mockingjays, she is used to communicate to the rebels, and it is often the song of others that she transmits. However, while mockingjays are used, they cannot be fully controlled or instrumentalised: They choose which melodies to pick up; sometimes they remain silent; at other times, they sing of their own accord (e.g. to announce a Hovercraft). And so does Katniss: She decides what to communicate and how to communicate it; she gives the call for rebellion a voice and meaning of her own (often defying her supervisors and friends), and she lifts her voice against the plans and strategies of the rebel soldiers when she feels that their military strategies deny the very humanity they are all fighting for.
So, if Katniss becomes the Mockingjay, she does not assume the power of Leadership. But this does not mean she has no power. Hers is the subtle but incalculable power of the communicative act, the symbol itself.