All right.
Attend the tale of Lucy Barker.
Benjamin's much-loved wife and Johanna's mother. Considered beautiful and virtuous. Lost her husband when he was found guilty of a false charge. 'Sat up there and sobbed by the hour' after his transportation. Raped by a judge. Tried to kill herself. 'Wound up in Bedlam instead', while her daughter was kidnapped by the man who raped her. Somehow, she got out.
Fifteen years later she's begging on the streets, and perhaps also a prostitute; this isn't made explicit in the film version. Talk about a tragic past.
She doesn't get a lot of love in fandom. There is one good reason for this - she doesn't have much screen time - and some not so good, which I'll let everybody decide for themselves. Given the lack of canon development, how do we know what kind of person she is? What does she actually do?
Lucy is a completely passive figure, which is probably one reason why Mrs Lovett, much more dynamic, is more popular. (Mrs Lovett is my favourite character in the film. She's close to being my favourite character overall - watch your back, Jack.) It's worth pointing out that this passivity was considered a desirable trait in Victorian women. Beautiful, virtuous and passive, Lucy is the ideal Victorian woman.
She is first seen in flashback with Benjamin, both of them enjoying each other's company and playing with their daughter. Clearly, they have a happy marriage. She doesn't get to react much when her husband is arrested, which is not a point against her, because there was nothing that she could do. She does look very distressed, and later 'sob[s] by the hour', so yes, she did care about what had happened to him. In both her first two scenes, she is shown taking care of Johanna. She was a loving wife and mother.
She resists the Judge's advances - passively; we don't hear any dialogue or reported conversation between them. All we see is that she refuses to 'come down from her tower' to him. But she doesn't accept him, either, even though without her husband, she is going to have to struggle for money - so she is not concerned with feathering her own nest. Mrs Lovett, on the other hand, thinks she should have accepted him ('had her chance for the moon on a string'). Lucy loved just as faithfully and far more selflessly than Mrs Lovett did.
Then Lucy goes to the Judge's house, at the Beadle's behest, thinking that Turpin might be 'contrite', and presumably hoping that he might bring Benjamin back. Instead she is plied with alcohol, which is likely drugged, and raped. Here she is utterly passive, going along with the Beadle's invitation and then being given no opportunity to resist the assault. Again - and this really, really should go without saying - this is not a point against her. She has no suspicions because she can't yet ascribe impure motives to anyone else, which indicates just how good-hearted she herself is.
"You will learn," Sweeney says to Anthony; and Lucy has. But unlike Sweeney, she doesn't turn her suffering on everyone else; she is still so good that even in her arsenic-addled, forced-prostitution state she tries to help people. She warns Anthony to stay away from Judge Turpin (not heeding Lucy's warning gets Anthony beaten up by the Beadle, mind, although in the end he was right in trying to save Johanna); is the only one who realises that there is something wrong with the pies and tries to tell everyone; and is even trying to warn Sweeney about Mrs Lovett the moment before she dies. Unfortunately she shared her husband's fatal flaw of not being a very perceptive judge of character.
She does have her moments of perception, though. Apart from her awareness that the pies are evil, she remembers Johanna and Judge Turpin (although it isn't clear to me whether she remembers their relation to her - does anyone have any thoughts on that?), and she recognises Sweeney before he recognises her, after fifteen years, and she hasn't any photographs for aide-memoires. Also, if she does remember the rape and circumstances surrounding it, it's very brave of her to follow the Beadle into the barber's shop.
So. Lucy is passive, but by the standard of her time, that is a good thing. She is also loving, loved, not a social climber or gold-digger (unlike Mrs Lovett), good-hearted, loyal, and, even when she has been dumped in the dregs of society with no hope of return, she tries to help others. And she is not involved with serial killing. What's not to like?