Sokka has plenty of reasons to hate his sister.
One of his parents' favourite stories had been from before Katara was born. While his mother was pregnant, they had tried to keep Sokka from becoming jealous by getting him excited about the new baby, his little brother or sister. It had worked a bit too well. They used to laugh as they recalled how Sokka used to ask, "When is my baby coming?" or "What is my baby going to be called?" She had belonged to him before she was even born.
His mother had such a kind smile, and when his father laughed, you couldn't help but want to laugh along with him. He had been too young to remember what life had been like before Katara was born himself, she had always been there. In one way, this kept him from disliking her too much, but in another, in made things worse. He was less of an individual, it seemed like sometimes his very identity was dependent on her. He could never be free of her.
It had been a very tense and excitement moment when Katara was born. Sokka and his father had had to leave their home and wait at his grandmother's while she and some women Sokka didn't know took over his house. It took much longer than Sokka could understand, and he soon lost interest in waiting, and started playing by himself instead on the floor of the igloo.
His parents' strategy hadn't worked perfectly. The new baby still annoyed him with her crying. He didn't like the way she made his parents tired, and it turned out that when he had been promised "a little brother or sister to play with," they had meant in a year or two. When she did come, his parents hadn't trusted him not to hurt her, and he found himself getting told off more often for being "too rough" with her, when all he wanted to do was be nice to her.
But gradually, Katara became old enough to play with, and to go outside. Sokka still got told off for making her cry, which he held against her because a boy wouldn't have cried so easily, but he also began hearing something new. He was told to look after her, keep her safe, make sure she didn't get hurt. Usually, he resented this. He didn't want to be anchored down by the little girl, he wanted to be free to run off with his own friends if he wanted to. But at the same time, he was proud. It made him feel old, mature, capable. Trusted.
And his little sister loved him. She absolutely adored him. She followed him everywhere, clinging to his hand and grinning ceaselessly at him with her tiny baby teeth. He was her hero, it was clear, and anything he did, she would imitate. Sokka soaked up the attention, encouraging her to worship him by showing her his boomerang. He could have shown her a snowball and she would have reacted with the same awe.
Those were the good times, before the bad times started. Before the Fire Nation raids. Suddenly, keeping his little sister safe meant more than keeping her away from the edge of the ice. It meant hiding behind snowdrifts, and not letting her hand slip from his as they ran. It was around that time that the snow would shift even though there was no wind when she cried against his chest, held in his childish arms as he looked over the snow to see if the coast was clear. Whenever stuff like that happened in front of his parents, they would always look at each other with such apprehension that Sokka knew it must mean something, though he didn't know what yet.
A lot of the women started being captured, hauled away from their homes and families and into the huge black metal ships that the Fire people came from. Some of the men were taken as well, but most of the fathers and husbands lost were defeated warriors who had tried to defend their tribe. Sokka wanted to join them. He knew what a warrior was at that age, and how important they were. His father was one, the best. They were the men that had the skills to defend their families, and Sokka had to look after his little sister, didn't he?
His father indulged him. He wanted his son to share in the culture of his people. He styled his hair and shaved the sides of Sokka's head, and carved a club for him. But he never intended for his young son to fight. Sokka found that out when he was a little older.
The people that were taken were the waterbenders. He heard his parents talking about it, and the war, when they thought he and Katara were asleep. He had seen them, of course, he had grown up around these people who could move the water and ice as they wanted. He had an aunt who could do it. He had wanted to be able to do it too, but he knew he couldn't. No one in his family could, or at least that was what he thought until he heard his parents mention Katara's name. His mother was talking about leaving, about taking her and Sokka and running, but his father said that would be cowardly, and that they shouldn't abandon their home and their friends.
Sokka was scared too now. If his sister was a waterbender, then they would want to take her as well, wouldn't they? Sokka made the decision there and then, without hesitation. If his sister was going to need a warrior, he would be it. He would have to shoulder the burden of the adorable girl-child, it would be his duty to make sure the smile never left her face.
But then, in a no time at all, he lost his mother. Katara had seen it happen and she couldn't stop crying. Sokka cried too, but he didn't understand. Why would the Fire Nation soldier take her from her home? They only took waterbenders. But when Katara told them what she had seen, their father was able to fill in the rest. He spoke in a shaking, hollow voice, his hands clenched into fists by his sides, his eyes a storm of anger and grief. Sokka's mother had claimed to be the last waterbender in the South pole, which they all knew was Katara. She cried the hardest, but Sokka stopped crying. He was angry now. Angry at Katara and her stupid tears. She had cost him his mother.
Days went by. Days and days. The snow turned white and smooth again, no more soot or running footprints. Life went on and Katara went unpunished for what she had caused, so Sokka pushed his feelings down and left them there. He was older now. His father began training him in earnest, showing him the perfect way to aim a boomerang, and teaching him how to apply the face paint of his tribe's warriors, making sure he understood the symbolism of every colour and every line. He learned that the wolf was a sacred animal, and that the warrior must ask for its strength and protection.
But then his father left, too. He raised an army of the remaining men and set sail to find the Fire Nation and fight against them for what they had done. Sokka had wanted to go, he had thought he was going. He needed to be a warrior too, he wanted to avenge his tribe. But his father made him stay behind. He was too young. He had to protect Katara.
And once again, it was all about her.
So he held his little sister as his father sailed away without him.
After that, life became almost too quiet. He trained everyday, and went fishing with Katara, who had grown into a bit of a nag. He occupied himself with the smaller children of the tribe, trying to pass on to them the warrior way of life their fathers had shared. It all felt so pointless and so small-time. He was having no effect there, he wasn't doing anything. Just looking after his little sister, who was needing him less and less everyday.
She kept trying to practice her water bending as if she didn't even remember why her tribe was so decimated. She flaunted it almost. Sokka's only consolation was that she wasn't making much progress with no one to teach her. Being able to stir a few ripples wouldn't make the Fire Nation return and snatch her away, he prayed. He couldn't bear to watch her do it though, shrugging her off whenever she tried to show him what she could do. He put her down, made fun of it, in the hopes that she would stop. She didn't, she just started insulting him back, something she never used to do before, and something that he got tired of pretty quickly. He knew she didn't understand all the sacrifices she had caused him to make, and he wasn't going to enlighten her. Keeping his own buried resentment of her hidden was part of his protection of her.
A part of him was jealous though. If he had a power like that, things would be different for him. He would be strong enough to make changes, improve his tribe's lives. But he didn't have the power, his sister, the victim, did. He had to train everyday, throwing his boomerang and swinging his club, in order to protect her, while she fumbled along with her power. He was the older one, he was the only defence the Southern water people had, but the spirits had gifted his little sister and left him high and dry. It just seemed like it was his destiny to struggle. Even so, Katara would always have him looking out for her, no matter what, he would never leave her side.
She had cost him both his parents. He wondered if she even realised it. At the same time, however, she had made herself the most precious thing in his world. She kept him from being alone. He owed his sorrow and his happiness to her. He was just afraid that if she ever got the water bending thing down, then that would end. She could be taken from him, or she could leave. She was outgrowing their decrepit remnants of a tribe more and more everyday. She had a vengeance in her heart just like he did. But maybe if he was lucky, she would take her older brother with her when she went.
Then they found Aang and everything fell down around them. The Fire Nation returned, but they had their ticket out. The siblings blew out of the South Pole on the Avatar's six-legged flying bison. Sokka was finally going to do something, finally going to make a difference. He was still the one who couldn't do anything, who couldn't bend water, air, earth or fire, but he never left his sister's side. He suffered pains of his own, more heartbreak added on to those of his childhood, but he protected his sister, even when she didn't want him to.
Sometimes she wouldn't listen to him. He knew he came off as the dumb one. But it wasn't his job to make her like him, it was his job to keep her safe, even from herself when necessary. So he would expose the Earth Kingdom freedom fighter, and he would even tackle the Avatar to the ground if he hurt her. And as for his own pain, he would suck it up as best he could.
When he realised that Katara was the person who came to mind when he tried to think of his mother, he wasn't too concerned. He would continue to honour her spirit, but he'd sooner have a sister alive than a mother dead.
And at the end of it all, they both had new people in their lives. He had Suki, and she had Aang. There was finally some peace in the world, and he didn't have to be a warrior anymore, though he now had the training to do a better job than he ever could have before. They were happy, but deep down he knew that all it would take was one word from Katara and he would leave it all behind him.
He just considered himself lucky that she didn't know that she held him in the palm of her hand.