Certainly this question is open to multiple interpretations, but I think one of the ways Scout could be considered intelligent is that she is very different from other children her age. As a child, she isn't typical. Her experience growing up in the home of a well-educated father, a black but respected housekeeper, and no mother, sets Scout up to be different from other children in the Jim Crow south.
Secondly, Scout has been taught to think by her father Atticus, who we all know is different from other adults in the town. He teaches her to see things from other people's perspectives, to walk around in others' skin, and to consider things from other angles. He taught her to read at a very young age, and speaks to her (and her brother) like they are adults. He explains everything with as much honesty as he can, and doesn't shelter his children from reality and truth.
All of these things combine to create in Scout a sense of worldliness, experience, and intelligence that, although she has never left home, is on a level of someone much older with far more experience.
Scout exhibits superior intelligence through her ability to read, her vocabulary, her understanding of the legal system, and her awareness. The trouble Scout has in school is highly ironic. A child of her intelligence should excel in the classroom, but instead her intelligence upsets her teacher. Scout learned to read before she attended school simply by sitting on Atticus's lap as he read legal journals and the newspaper. Her teacher, Miss Caroline, finds that annoying. On her first day of first grade, the teacher tests Scout and finds out she can read the entire reader plus the stock-market quotations from the newspaper.
At several points in the story, Scout displays a good understanding of the legal system, although it is somewhat tainted by her naivete. Atticus speaks to his children in adult words and encourages them to ask about anything they don't understand. In chapter 2 Scout asks Jem what an entailment is, and he gives a slang definition. Atticus explains the real meaning to her. She uses this knowledge to rescue Atticus from the mob at the jail in chapter 15. She starts talking to Mr. Cunningham about entailments, and it turns out to be this conversation that compels all the men to go home. During the trial, when Dill has to leave because he's so upset with the way the prosecutor treats Tom Robinson, Scout tries to explain to him how prosecutors need to act that way to do their jobs.
One indicator of Scout's intelligence is her inquisitiveness and her awareness. She takes in what is going on around her, and she learns a lot by listening to and observing adults. She asks questions to further her understanding. For example, she asks Calpurnia why her congregation sings hymns the way they do, and she asks Atticus what rape is. One of the most moving scenes in which Scout learns by being aware is on the day they learn of Tom Robinson's death. By observing Aunt Alexandra and listening to her reactions, Scout is able to rise to the occasion as her aunt does, using her "very best company manners" to serve the cookies to the missionary circle. That is the day she learns how to act like a lady.
Through her reading ability, vocabulary, legal knowledge, inquisitiveness, and awareness, Scout demonstrates that she is a highly intelligent child.