Signing the petition carries almost zero immediate pros. The petition is for general improved working conditions for the girls in the mills. The days are long, the air is painful to breathe, it's super noisy, the machines are quite dangerous, and the workers are given minimal breaks. The petition seeks to change all of that, but unless the petition actually works, all that it does for Lyddie is show the other girls that Lyddie is united with their feelings and willing to speak up about it.
The big con for Lyddie is that by signing the petition, she risks getting blacklisted from all of the factories. That means no work. No work means no paycheck, and earning money is important to Lyddie. She wants to be out from under the burden of debt that her family has put her under, and she wants the independence that financial success can bring her. If Lyddie is blacklisted, she has almost no means to attain her goals.