Studying ancient magic texts related to serpents might be a way. These texts could contain spells or incantations that can boost the power of snake summoning. Maybe there are also rituals that, when performed correctly, can expand the range of their ability, allowing them to summon snakes from greater distances. Additionally, they could learn from other magical creatures that have an affinity for snakes, like basilisks (if they can find a way to communicate safely), to gain more knowledge about snake magic.
His character might develop a more aloof side. Because he's raised by a snake, he may not be as open and friendly as in the original story. He could be more cautious and calculating in his actions.
It could be Nagini. Even though in the original series she was more of a servant to Voldemort, in the fanfiction, she could have a different personality and form a bond with Harry.
It could also be a snake from the Forbidden Forest. There are many unknown and mysterious creatures in there. A snake from that forest might be more in tune with the natural magic of Hogwarts. Harry, being the curious and kind - hearted person he is, might come across it during one of his adventures in the forest and start a friendship with it.
Perhaps the snake summoner is a descendant of an ancient wizarding family with a connection to snakes. Their ability draws the attention of Voldemort, who tries to recruit them. But the summoner resists and instead uses their snakes to gather intelligence on the Death Eaters and help the Order of the Phoenix. They could send snakes into Death Eater meetings to eavesdrop and relay the information back.
The snake's help could represent an unexpected alliance. In the Harry Potter series, snakes are often seen as symbols of Slytherin, which is usually at odds with Harry. So it's significant as it breaks the stereotype of enmity between Gryffindor (represented by Harry) and Slytherin.
The snake might use its ability to move stealthily. It could spy on the enemies for Harry Potter. For example, it can slither into hidden places where humans can't easily reach and gather important information.
Maybe it's an anonymous gift from someone who has been observing Harry. This person could be a former student of Hogwarts who has a special connection to snakes and sees that Harry has a kind heart and would take good care of the snake. So they leave the snake for Harry to find with a simple note that just says 'For Harry Potter'.
Snake magic in Harry Potter fanfiction can be used to show a character's dark side. Since snakes are often associated with Slytherin and dark magic in the original series, in fanfiction, a character using snake magic might be on a path towards the dark arts. For example, they could use snake magic to control others or gain power in a more underhanded way.
Some characters might develop their summoning abilities through intense practice. Just like Harry practiced his Patronus charm, they could spend hours every day trying to perfect the summoning spell, gradually increasing the range and complexity of what they can summon.
The snake might be significant in relation to the Slytherin house. Since snakes are associated with Slytherin, it could be a connection to that side of the wizarding world. It could be a symbol of Harry exploring the darker or more complex aspects of magic that are often associated with Slytherin, even though he's a Gryffindor.