Even for a superhero, Spider-Man has a particularly iconic origin, but one detail most people don't know is that the radioactive spider which bit him actually has a name. Peter Parker's origin has changed a little since his first appearance, keeping up with modern sci-fi ideas such that the spider is radioactive, bio-engineered, or magic depending on the version you read. However, its presence remains an essential part of Spider-Man lore to this day, and it even created other heroes such as Silk - another young hero who was bitten after Peter Parker.
Spider-Man's origin has two official aspects in Marvel lore. First, that a spider altered by advanced science gave him superpowers by biting him, imbuing him with arachnid qualities such as adhesive fingertips and enhanced agility. Second, that the spider was also an extension of the mystic Web of Life and Destiny, which finds a way to anoint a fitting Spider-Hero in every reality, choosing Peter because a lifetime of bullying meant he appreciated the responsibility inherent in becoming a 'predator.' In Marvel canon, magic and the mundane world go hand in hand, meaning neither of these explanations supersedes the other - they simply coexist.
It's therefore no surprise that the spider which bit Peter might be a little more complex that a household arachnid, and fans learn that it has its own life in Guardians Team-Up #5, from Andy Lanning, Andy Schmidt, and Gustavo Duarte. There, the Pet Avengers - a team of animal sidekicks from across the Marvel Universe - meet their opposite numbers, the Pets of Evil Masters (a play on Avengers villains the Masters of Evil.) Led by Doctor Doom's vulture, Vulture von Doom, the team includes some deep-cut comic references, such as Doom's attack tiger (from 1962's Fantastic Four #5) and Thori, the traitorous hellhound adopted by Loki and later owned by Thor. Also among their number is the spider which created Spider-Man, which is given the name "Bitey McSpidey Bite."
Is Bitey McSpidey-Bite Really Marvel Canon?
A memetic naming convention most famous from the real-life naming of the RRS Sir David Attenborough - which was the subject of a public campaign surrounding the suggested name 'Boaty McBoatface' - this name may feel a little silly to some fans, but there's every reason to believe its canon. The Pet Avengers are a canonical team, most recently helping Thor defeat his evil 'brother' Donald Blake, and there are many stories where Marvel has shown animals are able to communicate with each on a level humans can't understand, with a few 'intelligent' animals like Throg and Mr. Horse able to bridge the gap. Given the other stories in Guardians Team-Up appear to be canon, it seems that after creating Spider-Man, Bitey McSpidey-Bite went on to have its own life as a pint-sized villain. However, for those unhappy with the name, there is an alternative...
Marvel Also Calls the Spider 'Goddess'
In Marvel's Voices #1's 'Inspiration,' from James Monroe Iglehart and Ray-Anthony Height, fans meet 'Goddess.' Goddess relates a story in which after biting Spider-Man and Silk, she gained human intelligence and grew to enormous size. After being studied and named by a pair of scientists, Goddess eventually escaped out into the Marvel Universe, where she now acts as a dark vigilante, inspired by Earth's Spider-Heroes.
A More Sensible Alternative
As with many aspects of Marvel lore, these stories leave things open enough that fans can more or less pick the version they like best, knowing that the other simply took place in an alternate reality (or indeed imagine a world where Goddess eventually turns to evil as Bitey McSpidey-Bite.) For those who don't like either option, there's the Ultimate Universe (the reality where Miles Morales originated), where as Oscorp experiments, the spiders were simply numbered, with Peter bitten by Arachnid #00 and Miles by Arachnid #42. Whatever your preference, Marvel is seemingly clear on one thing - creating Spider-Man was just the start of this particular spider's bizarre adventures.