What is the significance of Mr Avery saying that bad children cause the unusual weather changes?
This is an interesting question. I do not think there is much significance in the words per se, but this is a transitional time in the book, which may be the main point. Here is the text:
For reasons unfathomable to the most experienced prophets in Maycomb County, autumn turned to winter that year. We had two weeks of the coldest weather since 1885, Atticus said. Mr. Avery said it was written on the Rosetta Stone that when children disobeyed their parents, smoked cigarettes and made war on each other, the seasons would change: Jem and I were burdened with the guilt of contributing to the aberrations of nature, thereby causing unhappiness to our neighbors and discomfort to ourselves.
Within in this context, Maycomb has a snowfall, which is out of place. This creates an uneasy setting. When Ms. Maudie's house burns down, this, too, adds to the ominous feeling and tone. A common literary technique in Western literature is to create something out of the ordinary to change the mood to a more ominous one. Lee does that here.
After chapter eight, the trial is about to take place. So, the childhood of Scout and Jem is about to end, and there will be a coming of age for both.
Of course, Jem and Scout do not know what is about to happen. So, they interpret the weather from the folk wisdom of Mr. Avery. However, the reader feels that something "bad" is about to take place. In a few chapters, the trial will take centerstage.
I thought the significance was that the children in Maycomb are often blamed becuase from others' point of view they don't know as much as adults and they do stupid things. However, the children should be respected more than the adults (well, some anyways) because they are not yet affectes by the ways of Maycomb and don't have certain issues like the grown ups (racism, being one example).