In Lyddie, how do family ties – or the lack of them – influence the characters' opinions about the efforts to improve working conditions,...

The girls whose families depend on them are less likely to sign the petition.

The petition for the ten-hour workday is dangerous for the factory workers.  Anyone who signs it is dismissed from her job and then blacklisted from the factory system so that she can never get another one. However, as working conditions worsen, the girls who do not have family members relying on them sign the petition and quit.

Lyddie in particular is...

The girls whose families depend on them are less likely to sign the petition.

The petition for the ten-hour workday is dangerous for the factory workers.  Anyone who signs it is dismissed from her job and then blacklisted from the factory system so that she can never get another one. However, as working conditions worsen, the girls who do not have family members relying on them sign the petition and quit.

Lyddie in particular is afraid to sign the petition.  She focuses on her family and her efforts to make money so that she can bring the family back together.  Lyddie needs the factory job, and can’t jeopardize her family even at the expense of the workers.

When the other girls ask Lyddie to go to a meeting for the ten-hour petition, she refuses.

"No!" Lyddie was startled herself that she had spoken so sharply. Both girls looked at her. "I mean, please, don't sign. I can't. I got to have the money. I got to pay the debts before‐" (Ch. 12)

The other girls have varying levels of commitment to others.  Betsy, for example, is putting her brother through college.  As long as he is in college, she can’t quit either.

"The golden lad finishes Harvard this spring. His fees are paid up, and I've got nearly the money I need now. My Latin is done. So as soon as I complete my botany course, I'll be ready to leave this insane asylum." (Ch. 13)

Lyddie’s only desire is to get her family back together, but Betsy’s feelings for her brother are not as affectionate as Lyddie’s.  Lyddie realizes that as the factory speeds up, more and more girls leave. The ones that leave are the ones who do not have families depending on the money they bring in.

Diana leaves when she finds out that she is having a baby.  She had been at the factory for about fifteen years and served as a mother-figure and teacher to the girls.  In a way, Diana left her "family" at the factory to have a family of her own.