So at the last of the movie Stone got a radio-connection from Earth. But unfortunately that wasn't from NASA. But from a mere man. Stone requested the man to make his dogs bark. Then Stone also barked with the dogs' sound. So what does it symbolize?
There are many things symbolised in the radio conversation with Aningaaq (an Inuit somewhere in the middle of an arctic wilderness). Some are more apparent if you watch the conversation from Aningaaq's side which, amazingly, was filmed as an add-on to the movie by Jonas Cuaron (son of the director and co-writer). The short movie can be seen here.
I'd say that the key theme from Stone's point of view is the balance between hope and despair. She has finally been able to contact someone, though they can't really communicate. But she does get to hear another voice before she is doomed to die. It may be her last chance to hear another person talk. There is a tension between the human contact and the possibility it will be her last human contact. During the conversation she encourages Aningaaq to make his dogs bark: she can't communicate effectively with the inuit, but dogs have a common language: she can howl at them, they can howl back. It's a little corny but it seems to work.
If you watch from Aningaaq's point of view another theme emerges. The original howl was from an old dog now in severe pain. The inuit is agonising about having to kill the dog as an act of mercy. Stone obviously can't understand this part of the conversation. But it adds a strange atmosphere to the short account from his point of view.
Stone also hears the sound of a baby crying, which she finds affecting. What the whole experience does to her mentality is unclear, but one possible role the scene plays is breaking her from a mood where she is resigned to her fate to one where the possibility of seeing earth again is worth fighting for. Even an incomprehensible muddled conversation is better than never having a conversation again.
Well, I would say she was just drifting off a bit at this point. She reached someone on the radio, just to realize that it was an unkown man that couldn't understand a word she was saying. In addition to this she was out of fuel and without any hope to get home again. She was at this point already preparing to die out there and was rather content with it.
So she was just happy to hear some human voice at the end of all hope (similar to user2176127's comment), be it just a remote man in Greenland (or wherever Aningaaq was from). So I'd say she was just joking around a bit with him and trying to make some simple conversation, maybe even to supress the thoughts about her serious situation and her supposed end. It was both a sign of her arrangement with her nearing end as well as of the desparation she probably wanted to hide with this little playing around.
So much to the reasons for her behaviour but I'm not sure there is really anything deeper to this situation (but am still looking forward to other possible answers).
Yes Matt, common language. Aninqaaq says his howling female dog is in pain, he implies that both he and the dog have accepted the dog must die. Both Astronaut Stone and the dog howl together, neither knowing the other is going to die, but speak the same forlorn howl. As stated above, Stone is comforted to be a part of life on earth even for a brief while. I think the lullaby facilitates Stone's death-wish but the baby cry foreshadows another rebirth for Ryan Stone.