I think that the book demonstrates the fickleness of children, and some adults, pretty well. For example, Tom faking his death is fickle. His decision to fight every boy who looks at him cross-eyed is fickle. In other words, he fights for fickle reasons and does not think before he acts.
First of all, I would point out that the second and last points above are not really examples of fickleness. Being drunk is not being fickle and trying to avoid punishment for murdering someone is not fickle.
I do agree that the other examples given are very good examples of fickleness in the book. We can also point out that Tom and his friends are fickle in their attitudes as they run away from home. First, they are eager to be "free" but then they get bored of it. In addition, Tom wants to hurt Aunt Polly by pretending to be dead. But he is also fickle to some extent because he wants to reassure her that he is actually alive so she won't feel too bad. These are both fairly normal reactions to the circumstances involved, but they can be interpreted as fickleness.
There are a number of incidents that occur in The Adventures of Tom Sawyer that demonstrate the fickle nature of many of the characters. Clearly, the townspeople are fickle. Tom forgets about Amy Lawrence, the girl he has kissed before, as soon as he sees Becky Thatcher.
Silas Dobson, the schoolmaster, is a drunk.
The townspeople have very few nice words for Tom or Huck until it is believed they drowned in the Mississippi River.
Nobody but Aunt Polly and the Widow Douglas have kind words for Huck Finn until he save Aunt Polly. Then everyone want to make him more civilized. Before that incident, Huck was shooed away.
Injun Joe conveniently placed the blame for Doc Robinson's murder squarely on Muff Potter's shoulders even though he was the one responsible for the murder.