While the main character's name isn't Gigabyte, the show features Bob "The Guardian", who gets sucked into the digital world of Mainframe every episode. He's tasked with protecting it from viruses and other threats.
While the show didn't explicitly use the term "internet," the digital world of Mainframe functioned very much like the internet, with interconnected networks, websites, and digital beings.
Let me know if you have any other cartoon mysteries you'd like to solve!