The family of Sister Lilith has been a member of The Order, and Catholics since generations. Yet her parents chose to name her after someone who was banished from heaven and is a completely negative (and even demonic) character in the religious mythology.
Was there any specific reason for the character to be named as Lilith?
According to Wikipedia, in the original comic book that the series is based on, Lilith actually is a demon (and her name is spelt "Lillith"):
Lillith, the Demon Princess, represents the kinder, gentler side of Hell. From her, we learn that there are many competing demon factions and not all obey the Devil.
...unlike her fellow demons, she carries no special grudge against the God who cast them out of Heaven or against His Church on Earth.
Aside from opposing Satan [...] Lillith seems to care for innocent life
The same article describes the Netflix series' version of Lilith as a "reimagining". Judging from your description (I haven't seen the series myself, or even heard of it until today), it sounds as though the Netflix series turned her into a regular human, but kept the name in order to establish a link with the original comics version of the character (and probably also for the irony of having a nun named "Lilith").
Lilith isn't the only character who was changed for the Netflix series: the main villain, a woman named Jillian Salvius, was a man named Julian in the comics.
