When the reader is first introduced to the novel's protagonist, her "name" is Brat. That name does not stick with her for very long, because Brat is taken in by Jane in order to be a cheap source of extra labor. Jane calls her "Beetle" because Jane found Brat/Beetle on a dung pile. The name Beetle stays in usage for quite some time. It is not until Jane breaks her ankle that a third name will come into usage.
Because of Jane's broken ankle, she can't travel. The solution is to send Beetle into town for supplies. It happens to be the Saint Swithin's Day Fair, so there are a lot of people and events going on with this particular supply outing. While at the fair, someone mistakes Beetle for somebody else, and he calls her Alyce. Beetle likes the name, so she decides to take it as her own.
This particular name change is symbolically important because it shows readers a confidence shift in Beetle/Alyce. She no longer feels like the weak girl that lived in piles of dung like a lowly bug. She is Alyce, the midwife's apprentice. Her knowledge of the craft has been growing. Her skill set has grown and improved. Consequently, Beetle's overall confidence in herself as a human being has grown. She isn't a bug, so she doesn't need an insect name. She is a human that can contribute to society, and the name Alyce reflects that new inner strength and confidence.
What a day! She had been winked at, complimented, given a gift, and now mistaken for the mysterious Alyce who could read. Did she then look like someone who could read? She leaned over and watched her face in the water again. "This face," she said, "could belong to someone who can read. And has curls. And could have a lover before nightfall."
"Alyce," she breathed. Alyce sounded clean and friendly and smart. You could love someone named Alyce. She looked back at the face in the water. "This then is me, Alyce." It was right.