I feel that, as Godzilla is the mythologizing of the experience of nuclear war, then the appropriate monster to mythologize our current age of the Internet is a serpent eating its own tail.
Alternatively, something like Nightgown Girl (aka Samara Morgan) from THE RING, except more spidery and serpentine, and able to come at you through anything you've connected to social media oh great now I need to sing my own damn self to sleep tonight thanks Kameron.