There is a speculation that Lady's death wake Bran up from his coma. What do you think about it?
I can understand anyone who subscribes to this theory, it's entirely respectable and it makes sense on its own.
I don't share it for two reasons:
The absurdity of the timeline.
The justification of life-for-life sacrifice.
Regarding the first point, we are given a specified timeframe between the Trident incident (Lady's death) and Tyrion's arrest at the Crossroads Inn. That's two sets of fortnights travelled on the Kingsroad in direct succession (First Ned, then Cat), four weeks. In those four weeks, Bran is supposed to have woken up, a raven dispatched to the Wall to inform Jon, Tyrion staying an additional day or two, Tyrion travelling all the way from the Wall to Winterfell, which took over three weeks one-way for on their way up, then trek down the kingsroad through the other half of the North, past the Neck and then the additional distance between the Neck and the Crossroads Inn. It's absurd to me. According to my own timeline calculations, Lady is killed around the time Tyrion arrives at Winterfell, giving him those four weeks to travel from Winterfell to the Crossroads. GRRM is no stranger to presenting chapters out of chronological order and I think it very much applies here. I don't judge anyone for disagreeing but that's how I read it.
Regarding the second point, it would give narrative justification to an absolute travesty of justice that shames every single adult involved. It would imply that Lady dying served a good cause. That Ned's failings here, the Cersei's cruelty, Robert's indifference, all of these things ultimately are good and necessary. I don't think that's probable, and I also think it's unnecessary. Bran had already magically survived with the help of the living direwolves and waking up is sufficiently explained by his inner decision to live:
He looked deep into the heart of winter, and then he cried out, afraid, and the heat of his tears burned on his cheeks. Now you know, the crow whispered as it sat on his shoulder. Now you know why you must live. “Why?” Bran said, not understanding, falling, falling. Because winter is coming. Bran looked at the crow on his shoulder, and the crow looked back. It had three eyes, and the third eye was full of a terrible knowledge. Bran looked down. There was nothing below him now but snow and cold and death, a frozen wasteland where jagged blue-white spires of ice waited to embrace him. They flew up at him like spears. He saw the bones of a thousand other dreamers impaled upon their points. He was desperately afraid. “Can a man still be brave if he’s afraid?” he heard his own voice saying, small and far away. And his father’s voice replied to him. “That is the only time a man can be brave.” Now, Bran, the crow urged. Choose. Fly or die. Death reached for him, screaming. Bran spread his arms and flew. Wings unseen drank the wind and filled and pulled him upward. The terrible needles of ice receded below him. The sky opened up above. Bran soared. It was better than climbing. It was better than anything. The world grew small beneath him. “I’m flying!” he cried out in delight. I’ve noticed, said the three-eyed crow. It took to the air, flapping its wings in his face, slowing him, blinding him. He faltered in the air as its pinions beat against his cheeks. Its beak stabbed at him fiercely, and Bran felt a sudden blinding pain in the middle of his forehead, between his eyes. “What are you doing?” he shrieked. The crow opened its beak and cawed at him, a shrill scream of fear, and the grey mists shuddered and swirled around him and ripped away like a veil, and he saw that the crow was really a woman, a serving woman with long black hair, and he knew her from somewhere, from Winterfell, yes, that was it, he remembered her now, and then he realized that he was in Winterfell, in a bed high in some chilly tower room, and the blackhaired woman dropped a basin of water to shatter on the floor and ran down the steps, shouting, “He’s awake, he’s awake, he’s awake.”
Bran wakes up because he chooses to wake up, even knowing - if subconsciously - that it means serving a specific, scary purpose.