My daughter had the same problem. When she was about 9, I was looking in her mouth and discovered a gross, white cyst of repulsive-smelling stuff that had built up in one tonsil. It's called cryptic tonsils. There is a slit in the tonsils that fills up with smegma, essentially. Or, smaller white cysts can just build up ON the tonsils. I spent months pushing the goo out of her tonsils with a long q-tip, and as she has matured her tonsils have shrunk and don't fill up much anymore.
By the way, the hard, white ball that Camille's daughter coughed up (below) was actually one of the things that builds up on the tonsil. It's called a tonsil stone or tonsiloth.
When I discovered this, I checked my other family members, and one son had the slits while the other did not.
If your daughter will let you, take a good look at her tonsils and I bet you will find she has it. If you have a strong constitution, you can put 'cryptic tonsils' on YouTube and watch someone getting theirs professionally cleaned.