Depends how close to canon your Lolth is, who's in the party, what do they know, etc...
For example, if anyone in the party is a drow who is well aware of, or has experienced first-hand, the kind of abuse and total misery that Lolth forces on the drow; if they know that she's a sadistic deity whose only goal is to keep the drow enslaved to her, suffering and crawling at her feet, then that character might not give a flying that Lolth has a family, and even choose to personally end her. After all, Lolth knows how to put up a facade, and all she does 24/7 is thinking how to backstab others, how to cause strife, or how to gain more power. If the character is aware of this, as compassionate as they might be, they will find it seriously hard to not hunt her down, because that would mean condemining thousands to decades, if not centuries or millennia, of pain. Sadly, this is the level of caricature-evil that canon Lolth is--the kind of evil that has 0 redeeming qualities.
A character who hasn't experienced first hand any of her cruelty and only knows her by reputation, might choose to help her instead, because rumors are only rumors, and even though people like to paint their enemies as 100% evil and wrong, the truth might be different.