I don't think the question is unreasonable, ignorant or stupid.. but society isn't as reasonable as it used to be. I also understand the desire to discipline your child, which is why I suggested a way to show that stealing is wrong without exposing your son to the possibility of ridiculous consequences.
Back when I was a kid, parents would often drag an embarrassed kid back into a store to return a stolen item. The kid was suitably humiliated and learned a lesson. I grew up far from being "a very privileged person," and my family only knew the criminal justice system as something that protected us from very bad people.
Unfortunately, it's a different world today, and many parents don't understand these old-school tactics don't fly in an increasingly hysterical world. Our country imprisons a larger percentage of its population than any other Western nation, but politicians are constantly promising to pass tougher laws that will lock up more people for longer sentences.
When I was in elementary school, every second grade boy used his Big Chief tablet and extra thick pencil to draw battle scenes of stick-figure army men shooting other, with jet fighters dropping bombs and strafing the enemy, and explosions all over the place. In the post-Columbine era, kids who do drawings like that wind up going to the school counselor, then to a psychiatrist for a Ritalin prescription.
Bottom line: getting outside agencies involved in disciplining your children is a bad idea. Once you get the police or a store manager involved, you lose control of the outcome.
There's no way to predict what might happen. It depends on the type of store, the item taken, your child's age and the store manager's mood.
For instance, if your son is eight years old, the store manager might think it was cute; if he's sixteen, they might call the police to take a report. (The store cannot "charge" him -- that's up to the District Attorney.) The owners of a Mom & Pop corner grocery store may be less likely to turn it into a big deal than the manager of a big-box store... but Mom & Pop might also be hardheads who want to make an example out of your kid by calling the cops.
There's also the chance they will take your information and turn it over to a law firm that will try to bully you into paying hundreds of dollars. See the link below to my legal guide on Civil Demand Letters.
If it was my child, I would send the item back to the store anonymously, and make the kid volunteer at a homeless shelter or, if he's too young, make him give up a favorite toy to charity.
Please understand that this is a general discussion of legal principles by a California lawyer and does not create an attorney/client relationship. It's impossible to give detailed, accurate advice based on a few sentences on a website (and you shouldn't provide too much specific information about your legal matter on a public forum like this site, anyway). You should always seek advice from an attorney licensed in your jurisdiction who can give you an informed opinion after reviewing all of the relevant information.
If it were my child, since Mr. Marshall raises that subject, I would hire a lawyer to defend him or her and stand behind the child no matter what. Forcing your own flesh and blood to turn himself in is about the equivalent of ratting him out. Normal families don't operate that way. I realize you live in Palo Alto so you are almost certainly a very privileged person and know nothing about the criminal justice system.
Even so putting your child in harm's way to teach it a lesson is about the dumbest idea I've ever heard. I don't mean to insult you and certainly wish the best for your child who apparently has as much to fear from his parents as from the police but in our curt system there are two pleas: guilty and not guilty. There is no provision for apologies and making one will constitute an admission of wrongdoing that may have collateral consequences even if your child is a juvenile.