They are arguing over whether or not to sign a petition calling for improvements in pay and conditions. The Concord Corporation has been speeding up the looms again. Lyddie is not too worried, as it gives her an opportunity to make more money, but some of the other girls at the factory cannot handle the pace. On the whole, the acceleration of production is taking its toll on the physical health of the workforce. Amelia is so tired and crabby that she gets into a silly argument with Lyddie when she will not put her book down and take a short walk by the river.

Betsy has half a mind to sign the petition, but Amelia is none too enthusiastic about it. Lyddie does not say anything, but she is concerned that if the petition succeeds, then the girls will end up working fewer hours. If they work fewer hours, they will earn less pay. Lyddie has been working hard to pay off her family's debts, and she is concerned that working fewer hours at the factory will make it harder for her to do this.

The argument that your question is asking about occurs in chapter 12 of the book.  There are some minor arguments that occur during this chapter, but the main argument among the three girls deals with the factory and the present working conditions.  Betsy suggests a couple of options.  One option is for the girls to stage a walk out to force better working conditions.  The other option that she is thinking about is signing the petition.  Betsy and Amelia argue over the pros and cons of those ideas, and Lyddie is opposed to both.  At this point in the story, Lyddie is working multiple looms and making a fair amount of money.  A walk out means that she is out of work and a paycheck.  If she signs the petition, she might be black listed from all of the mills.  Again, that would mean no money.