In alluding to the Rosetta Stone, Mr. Avery seems to be making a joke at the children's expense. Perhaps he thinks they will ask him what this stone is. (If so, he doesn't know Scout very well.) While the phrase "written in stone" is commonly used to mean something mandatory...
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or undeniable, such as on Moses' tablets, Mr. Avery adds a twist. He names a particular stone, which he knows to be very old, to emphasize the inflexibility of his source.
The irony is that Scout, who is always reluctant to admit to childish ignorance, seizes on the content—the weather—and imagines that he had referred to a meteorological authority. In this allusion, Harper Lee is gently poking fun at both the grown up and the girl.
Apart from the fact that Mr. Avery quite incidentally got the weather correct, the real irony is that both Mr. Avery and Scout did not know anything about the Rosetta Stone. Here is Scout's take on Mr. Avery.
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.
The Rosetta Stone is a text written in three languages (hieroglyphic, demotic, and Greek) by a group of priests in Egypt to honor the Egyptian pharaoh. It lists the achievements of the pharaoh.
The irony is that Mr. Avery can use something scholarly to persuade children that he can preternaturally predict the weather and the behavior of children, when he does not understand something himself.
The other irony is that in honor of Mr. Avery (or to mock him), Jem and Scout make a snowman. Atticus eventually yells at the children for mocking a neighbor.