Before I do so, there are a few minor points that might be worth noting. The first is that even in a jury system, juries are sometimes a much smaller part of the actual operation of the justice system than you might think. Most criminal cases in the U.S., for instance, do not use juries because they do not go to trial -- the accused might plead guilty for instance, or waive the right to a jury trial. The second is that not all jury systems are the same. The U.S., for instance, has a dual jury system. Not only does it have a petit jury, which is used in trials; it also has a grand jury, which investigates indictments. Many of the complaints against jury trials (e.g., that jurors don't have much room to investigate the mattter on their own) are false, misguided, or confused if they are taken to give reasons against grand juries. In what follows I will be considering only the petit jury (jury trials), and primarily focusing on the U.S.; but it's worth keeping in mind that there are many different possible uses of juries, and not all of them can be criticized on the same principles (although the defense of these various uses often relies on the same principles for each).
So here is the argument, in brief form; it's possible to get more elaborate in the details, but I'll be as concise as possible.
(1) Contrary to a belief that seems very common among jury opponents, it is the primary purpose of the court to get to the truth of the matter, not the primary purpose of the jury. While the case against a jury's ability to pinpoint the truth is often exaggerated, it is nonetheless true that the primary point of adding a jury to a court is not to get to the truth but to protect the people. As Scalia very neatly put it in the Blakely decision (PDF):
Our commitment to Apprendi [a prior precedent affirming the right to jury trial] in this context reflects not just respect for longstanding precedent, but the need to give intelligible content to the right of jury trial. That right is no mere procedural formality, but a fundamental reservation of power in our constitutional structure. Just as suffrage ensures the people’s ultimate control in the legislative and executive branches, jury trial is meant to ensure their control in the judiciary. See Letter XV by the Federal Farmer (Jan. 18, 1788), reprinted in 2 The Complete Anti-Federalist 315, 320 (H. Storing ed. 1981) (describing the jury as "secur[ing] to the people at large, their just and rightful controul in the judicial department"); John Adams, Diary Entry (Feb. 12, 1771), reprinted in 2 Works of John Adams 252, 253 (C. Adams ed. 1850) ("[T]he common people, should have as complete a control . . . in every judgment of a court of judicature" as in the legislature); Letter from Thomas Jefferson to the Abbé Arnoux (July 19, 1789), reprinted in 15 Papers of Thomas Jefferson 282, 283 (J. Boyd ed. 1958) ("Were I called upon to decide whether the people had best be omitted in the Legislative or Judiciary department, I would say it is better to leave them out of the Legislative"); Jones v. United States, 526 U. S. 227, 244–248 (1999). Apprendi carries out this design by ensuring that the judge’s authority to sentence derives wholly from the jury’s verdict. Without that restriction, the jury would not exercise the control that the Framers intended.
While this analogy (jury : judiciary :: suffrage : legislature) is more complicated than it looks, it will play an important role in almost any serious defense of the jury system. As Scalia puts it, the jury system is a "circuit breaker in the State's machinery of justice". We will discuss this point a bit more when we get to nullification, below. Right now the point to keep in mind is that the jury serves to give the people some measure of "just and rightful control" over the judicial process in cases that most clearly concern them. It's this attitude, no doubt, that led the Founders to place deprivation of the right to trial by jury as a serious complaint that contributed to the case for independence.
What are some of the ways in which a functioning jury system can help to protect the people? First, of all, the jury system helps to provide a defense against conviction on a weak case -- thus it's common to require a jury's unanimity in a criminal case, and to allow the possibility of a hung jury. Even the fact that a jury is composed of peers, broadly speaking, serves to some degree as a protection, since it limits the degree to which you can be dominated by bureaucrats and magistrates alone. Further, it has become clear over time that juries provide some protection of citizens against laws that are widely considered bogus or difficult to understand. It is necessary to convince the jury, composed of ordinary people, that a genuine crime has been committed; and some juries have gone so far as to ignore the law, declaring innocent people who are proven to have violated the law. This type of case has itself been criticized, and it can certainly be abused, but it is important to note that it is very limited: juries are almost always reluctant to ignore the law altogether, and the authority of a jury applies only to a particular case, and so is not a general nullification of law. Thus the jury system can allow for extraordinary mercy in extraordinary situations; it can moderate laws, verdicts, and punishments that would commonly be considered excessively harsh; and it can give people an additional measure of defense against injustice in the system.
More than this, however, the existence of juries legitimizes the criminal justice as an instrument of the people. Juries act as observers of the judicial process; they are a way in which citizens can actually participate in the justice system; they guarantee that there is in the system a body of ordinary people to whom one can appeal for sympathy. They add another level of conscience to the attempt to maintain justice. Further, they assist the judiciary by diffusing responsibility: it is not merely the judges who are responsible for the acts of the justice system; the people are represented in it as well. Of course, most of this function of juries is only possible in societies willing to put a high degree of trust in juries. Without such a trust, juries will still legitimize the system, but only in a minor way. Where such a trust exists, however, it will be virtually impossible to convince people that the justice system is legitimate without a fundamental use of juries.
The most common complaint against juries, I suspect, is the complaint that they are defective means for finding out the truth. It is true that they are uneven in this regard. But their defectiveness is often exaggerated. Juries can often improve the judicial decision process by bringing a wide range of diverse backgrounds into contact with the case, by making sure that the decision is arrived at in a way that is accountable to collective recall, and by being a very explicit form of conscious scrutiny. It is true that jurors are often confused by obscure laws or unexpected facts, and often swayed by rhetoric alone; but whatever problems there may be on this front, they certainly don't outweigh the benefits that both sides of the case and society at large are able to gain from the existence of the jury.
None of this eliminates the possibility that there might be ways to improve such jury systems as exist. But it does provide a strong presumptive argument, I think, for taking the jury system seriously as a key part of free and popular government. Attempts to abolish or limit jury trials are attempts to abolish or limit the rights and protections of the people, and should only be tolerated if excellent reasons can be brought forward for them.
[ADDED LATER: As a commenter points out, I need to clarify that the argument laid out here is based only on consideration of criminal trials, since I'm primarily concerned with people who suggest that there is a problem with juries, simply speaking. A number of the points don't contribute much toward a defense of the use of juries in civil trials. In such trials the 'protection of the people' defense is far more limited and the legitimization defense is far less impressive. I think one argument that remains in full force is the argument that forcing the justice system to face the fact that law needs to be reasonably intelligible to ordinary persons even in matters of civil law; but it's not nearly as obvious that juries are a good way to do this in civil matters.]
