The Ability to "bend" sound. Contains LoK B3 spoilers.
This is a new idea I've had, something that has not been directly represented in the show, but has been shown evidence of possibly occuring and on this website has been partly explored (Elegys, thunderbending).
"Soundbending" is not exactly the ability to bend sound, but rather the ability to manipulate the medium it moves through, namely, air. Soundbending is not really the ability to create specific sounds, but rather the ability to stop sounds. I will provide examples of how this works and how it is justified.
The basic ability of soundbending is to mask the sound of one's movement. My idea of how this is achievable is that an airbender creates a vacuum around their body, but preserve themselves in a pocket of air that fits their body and allows themselves to breath. This vacuum does not extend far, but far enough to prevent all soundwaves from entering or exiting the vacuum. No one can hear the airbender move or speak, but the airbender can not hear anything aside from the sound of his own body. This is justified somewhat in LoK by Zaheer's ability to asphyxiate. This shows that airbenders are capable of completely removing the air from a given area, and thus could theoretically create a vacuum where needed. This vacuum would not be visible, and here's why. In the show, airbenders were visibly shown to airbend, but this can be interpreted as a method of explanation for what they were doing, as you wouldn't literally see someone airbend obviously.
Obviously the airbender would be able to keep him or herself alive, due to this example. Imagine the vacuum as water, a representation of where you can't breath. Aang was frequently shown to create air bubbles in water, thus proving he could create them in a vacuum, since he created the vacuum and could keep some air for himself. The only inconsistency of this example is the size of a body of water compared to the size of the vacuum. The vacuum would be only an inch or two thick, enough to stop sound. Unfortunately for the airbender, he or she would remain detectable to someone with seismic sense, unless the airbender also possessed the power of flight.
This ability to soundbend also opens up the possibility of sound tunnels, for close, silent communication. Two or more airbenders create the vacuum around their head, and also create tunnels of air surrounded by vacuum, through which they can communicate, unheard to anyone else. In addition they could create sound bubbles, to hold silent conferences or to eliminate the loud sound of warfare, machines, etc.