It's actually a perfect name choice to embody their worldview.
Say the word aloud. It is coronal heavy, as in your tongue curls and twists suggestively as you say it. It's a soft, gentle word and yet it intones sinisterness, an ancient mystery. It suggests everything that they aspire to be - seductive, powerful, but importantly, complex - misunderstood, underestimated. To say the na,e Lilith is to invoke images of darkness and lost secrets, hidden depths and enigmatic whispers in the night, a pall of uncertainty that tugs at your soul and threatens to pierce the unfathomable veil of the realm corporeal NO MOM IT'S NOT A PHASE
It's also, of course, a subtle diss toward women - lesbian...shipping aside, what appears to have happened is that Lilith was originally mentioned once in the Old Testament as a beautiful demoness with manipulative powers who thrived on chaos, but in a Middle Ages 'reimagining', called the Alphabet of Ben Sira (which might literally be a
giant troll, no joke), she was originally molded of the same clay as Adam, which made them equals and would make Lilith the first woman, but Lilith refused Adam's dominion over her, so she fled Eden into the desert. Eve was created from Adam's "rib", which may be a mistranslation - the implication either way is unlike Lilith, Eve is Adam's 'lesser'. Part of the complication is that Lilith's evil is the result of tragedy - God commands her to return to Eden, and for every day that she remains in exile, one hundred of her children (which she apparently spawns like insects) will die. Lilith believes herself to be betrayed by Adam (cause she's a woman) and defies God by uttering His sacred name and resolves to retaliate by murdering as many of God's children (mortal babies) as she can unless they are warded with her name. In this way, trapped by her very nature, Lilith critically maintains the balance of good and evil in the world - a.k.a, literally the most important character all along.
The reason Lilith's expanded universe origin story might be a troll is because she is explicitly used in the Alphabet as a feminist foil for God and Adam - Lilith is driven to vile behavior through her instincts as a mother, and her bitterness toward Adam is a commentary on men and women being unable to tolerate each other's agency. The rest of it picks at holes or asks questions about the whole creation story.
Basically, she personifies their headcanon views of themselves - misunderstood, mistreated outcasts who turn to dark methods or do despicable things because they were backed into a corner by ruthless powers - except it might secretly all be a massive pisstake and it was just a modernist byproduct. How perfect is that?