A lot, but let's be practical.
Or nothing, depending on how it is taught. I experienced the joy of obligatory 4th grade Spanish instruction, where there were virtually no Spanish-speaking teachers (they relied on a edu-tv show): it was not at all effective and didn't last a year. Teacher training on the subject matter is a big problem: so it should be taught like chemistry or high school health, and not mathematics or literature (focused class taught by an expert, for juniors or seniors). There used to be "high school civics" which taught a little bit of this, but primarily focused on politics and not the consequences of politics.
Before constructing a curriculum, you need a goal (well-articulated and reasoned, not just an emotional position like "it would be good to know this"). History and literature are more on the side of "that's what it means to be educated", whereas mathematics and grouse-trapping is more on the side of "survival basics". Elementary legal education is more on the side of survival skills, which means, given a choice of an hour on voir dire versus an hour on 4th Amendment search and seizure, the hour should be spent on search and seizure.
Although it is a bit abstract, elementary jurisprudence is one of those mixed survival-skill + abstract fundamentals that is so important that everybody should understand (some of) it. The reason is that it goes to the question "what is the law?". Most people believe incorrectly that "the law" is only that which was passed by Congress / the legislature, and there is very little appreciation for the necessity of interpreting the words of the law-givers. The concrete target of elementary jurisprudential education should be an understanding of why we have appellate courts.
Contract-reading would be rather high on the list of priorities, at least as long as attorneys are allowed to charge for their services and free legal advice is not deemed to be a fundamental constitutional entitlement. It is easy to say "you should have your attorney read that contract", but very few people do. The goal is to improve people's ability to understand the consequence of contracts so that they don't mindlessly agree to everything, given the reality that people are not going to take all of their contracts to a lawyer and ask if it's okay to sign. Every citizen should know when they are in over their heads and should hire a lawyer.
Also high on the list would be a solid understanding of "my rights as a citizen". People tend to intuit what the law is in terms of their feelings about "my rights", so if you feel that you have a right to barbecue a hamburger, then you will tend to think that it is legal to do so. It is therefore very important that every citizen have a solid understanding of what "your rights" are, and ways in which your feelings can be mistaken. Basic education on the Commerce Clause (and state relatives) is very important, given that the Commerce Clause is a major source of counterexamples to people's intuitions about their rights.
