In certain 'Harry kills the dragon' fanfictions, Harry could be given a special magical weapon specifically designed to slay dragons. He might have to go through a series of trials to figure out how to use this weapon effectively. It could be that the weapon has to be powered by his own magic and emotions. Once he masters this, he is able to take on the dragon and kill it. For example, the weapon might need to be charged with his feelings of courage and determination, which are strong in Harry, and then he uses it to strike a fatal blow to the dragon.
Well, it often starts with an unexpected encounter. Maybe Harry stumbles upon the dragon in a hidden place. At first, the dragon might be wary but then realizes Harry is special. As time goes on, they start to trust each other. The dragon might start to protect Harry from threats, and Harry in turn, might learn to communicate with the dragon in a unique way.
One consequence could be that he becomes a hero among his peers. Everyone in Hogwarts might look up to him even more. The other students could see him as a real dragon - slayer, which gives him more popularity and respect.
It could be Norbert, the dragon Hagrid once had. Since Norbert was a young dragon when we first saw it in the original story, it might have grown up and have a lot of stories to tell Harry.
In many fanfics, Harry being a Dragon Animagus can bring about really cool storylines. For example, it could add a whole new layer of power to his character. He might be able to use the strength and abilities of a dragon in battles against Voldemort and his Death Eaters. It also could change his relationships with other characters. Some might be in awe of his new form, while others could be afraid.
The significance could be that the egg represents new life and potential. In the wizarding world, magical creatures like dragons are powerful and mysterious. The egg might be a symbol of a new adventure for Harry. It could also be a sign of trust from the dragon, as it's entrusting Harry with something precious.
In many fanfictions, Harry might use his powerful spells like the Stunning Spell or the Killing Curse if the story allows for such dark magic use. He could also use his intelligence to outwit the dragon, perhaps finding a weakness in its armor or behavior pattern and then using a well - timed spell to defeat it.
It could be that the dragon was an egg he found by accident. He, being curious and having a soft spot for magical creatures, took the egg and hatched it, thus becoming the owner of the dragon. In the wizarding world, there are often unclaimed or lost magical items and creatures, and Harry might have just stumbled upon this dragon egg.
There's also a possibility that it's a Chinese Fireball. These dragons are very beautiful and powerful. In a fanfiction, the author might choose this type of dragon for Harry Potter to own because it would add an element of exoticism and grandeur to the story.
It could be an entirely new type of dragon created specifically for this fanfiction. The author might have wanted to introduce a unique dragon with special abilities, like a dragon that can change its color to blend in with its surroundings or one that can breathe not just fire but also some kind of magical ice. Harry would then have an even greater challenge when trying to kill it with the sword.