The story
In the early 2000s, a fan going by the pseudonym Severitus (presumably constructed from Severus + -itis) issued a challenge requesting fanfics with several specific elements, including Snape being revealed as Harry's true father. This became known as the Severitus Challenge, which was later shortened to Severitus, and came to refer exclusively to the paternity trope.
Etymology from Wiktionary, CC BY-SA 4.0.