This narrative quotation by Scout is a comment about her Aunt Alexandra's obsession with her own family's heritage. She believes that the Finches are descended from the finest of all of Maycomb's families, and the quote simply means that Alexandra loves to make unflattering remarks and comparisons about the other families in Maycomb. For example, all of Maycomb's families had some kind of Streak--except for the Finches:
A Drinking Streak, a Gambling Streak, a Mean Streak, a Funny Streak.
According to Scout, Aunt Alexandra believed that Fine Folks were best judged by how long
... a family had been squatting on one patch of land...
And since Simon Finch was the earliest inhabitant of the area, the Finches were the Finest Folks of them all.