Going by Lilah's overall behavior and attitudes, I doubt it.
However, Lilah does care for her own mother, and perhaps she could relate to something in Bethany (but this would not overcome Lilah doing her job who helped put that girl through hell). Makes me wonder of Lilah's backstory before she came to Wolfram & Hart...
I did not think Lilah genuinely cared for her; she's just really good at her job. She has fantastic people skills for a queen bitch. And dealing with people with low self esteem and problems that seem impossible to solve is like taking candy from a baby for her.
Despite that, it's clear that Lilah has a lot more of a heart than what she shows. I just don't think this particularly was an example of that.
Does anyone think Lilah generally wanted to help Bethany?
Lilah doesn't help.
She hired men to attempt to rape Bethany in an alleyway, and then brought her rapist father in to force a showdown. She refers to bringing the dad in as 'pulling the trigger'. She wanted an assassin for the firm. She wanted Bethany to learn to control her power in order to be an assassin. She did not care how much she had to hurt Bethany in order to teach her that control. Her manipulation and faux care for Bethany when she is in her house is frightening to watch, because it is so expertly done. But the fake naive 'there's a power about you, Bethany', and the way she stands watching Bethany have nightmares, with a big smirk on her face, absolutely tell us that her 'care' is pure manipulation.