Note: This isn't about the Hazbin Hotel character, but it can connect to the show. I did a philosophical analysis on Lucifer like half a year ago and I like to come back to the actual characters that inspire HH and HB. Skip to the positive and negative sections at the bottom if you don't wanna read all the fluff.
Everyone on this wiki should probably already know, but Lilith is believed to be Adam's first wife instead of Eve. Instead of submitting to Adam, she believed she should have a more dominant role in the relationship, but was banished from the Garden of Eden for her ideas and beliefs. She's now a symbol of feminine autonomy, being associated with succubi, which prey on men.
Another key detail about her is that she often preys on infants and pregnant women, causing them (the women) to have gruesome and difficult pregnancies, often ending in death.
History: Originally, Lilith wasn't a part of the classical Christian doctrine at all. Her story traces back to pagan customs, and were originally used to scare women into submission. Her story wasn't even technically a part of Jewish doctrine, either, even though a lot of people like to say it is. A lot of different priests and cultural groups disagreed with the story for a variety of reasons.
Negatives: I don't really like Lilith a lot because it's hard to discern the message behind her existence. On one hand, I don't understand why so many people value her as a symbol of feminism. Standing up for your own rights against a man takes strength and courage, but people ignore what she did after that. Feminism is the belief that women deserve the rights to be their own aspiring individuals -- Lilith is now the symbol of a group that preys on men and vulnerable women and children. Suffocating pregnant women is... feminism? There are so many other women in mythology that fought for their own rights and didn't go batsh*t on uninvolved mortals -- although rare, they're a better symbol of strength.
Positives: It's important to consider that Lilith's story was originally meant to diminish the rights of independent women. She could've been made out to be a gruesome monster because a bunch of priests wanted to say that independent women are "dangerous" and only seek to hurt others for personal benefit. In fact, this is a VERY likely story.
It's hard to develop a concrete opinion of her representation because we'll really never know for sure.