He might choose holly. In the original series, Harry's wand has a holly wood. Since he's making his own in this fanfiction, he could still be drawn to it for its traditional associations with protection and courage.
He might choose a phoenix feather because it represents rebirth and magic. And for the wood, perhaps he'd pick holly, which is often associated with protection in the wizarding world.
In fanfiction, his wand could potentially be used to reveal secrets. Snape was full of secrets himself. Maybe his wand could be a key to unlocking hidden chambers or revealing long - lost knowledge. It could be used to decrypt old spells or manuscripts that were crucial to the wizarding world's history.
Well, he starts by going into the Forbidden Forest. There, he communes with the trees, seeking one that calls out to him. Once he finds the perfect tree, he uses his magic to gently remove a branch. He then spends time carving and molding the wood, all the while focusing his magic into it. For the core, he might go on a quest to find a special magical creature from which to obtain a suitable element. After getting the core, he combines it with the wood in a ritualistic way, using his most powerful spells to bind them together.
One scenario could be Snape using his wand to communicate with the dead. Maybe he discovers a spell that allows him to use his wand as a conduit to speak to Lily Potter. This would add a whole new layer to his character, as his love for Lily was a major driving force in his life.
Perhaps it was related to some new and untamed magical ability within Harry. For example, if he discovered a hidden power within himself that was too much for his wand to handle. His wand, which was made for normal magical use, couldn't withstand the intensity of this new power, and thus shattered.
He might have lost his main means of casting spells. This would put him at a great disadvantage in any magical battles or challenges. Without his wand, he would have to rely on other forms of magic or his natural abilities, which could be much more difficult.
In Harry Potter fanfiction, a character being 'his own man' might mean that he has the independence to make his own choices, regardless of the expectations of others. For example, a character might not follow the typical path set by the original story, like a Slytherin who decides to oppose the pure - blood supremacist views of his family.
In fanfiction, if Harry Potter snaps his wand, he might lose a great deal of his magical power. His wand is his primary tool for casting spells, so without it, he'd be in a very vulnerable position. He could start to explore alternative forms of magic or seek out a new wand, perhaps through some extraordinary adventure. This could also lead to him relying more on his friends and the skills he has developed outside of wand - based magic.