This is a really neat writing assignment. You could have a lot of fun with it, depending on which character you pick. Remember, no matter which character you pick, write from the first-person perspective. If you want to make this assignment easier on yourself, you might want to consider writing from Brigid's perspective. She and Lyddie become quite close, so there is a lot of source material there. I would recommend picking Brigid's first day on the factory floor as the incident to write about. Her experience of the first day is going to be very similar to Lyddie's. The noise and pace of things have to be completely overwhelming to Brigid, like they were to Lyddie, so you could look to the chapter that narrates Lyddie's first day for some inspiration.
It would then be fun to explore what Brigid's thoughts of Lyddie are. We know that Lyddie first grows to resent Brigid's seeming incompetence at working the machines. Helping Brigid also means that Lyddie's own output is slowed. We know Lyddie isn't happy about the situation, but how does Brigid feel about Lyddie and her help?
Another fun sequence to explore would be the short sequence where Lyddie confronts Mr. Marsden and blackmails him with the letter about his advances on the factory girls. You could write him as dumbfounded, enraged, or a combination of the two.