I agree with the person above who said the problem with all political systems is that they are run by people. I would say that a constitutional monarchy, conceived in such a way as to limit the monarch's power to harm the state but enhance the monarch's power to do good to the state, minimizes this problem.
The trouble with republics is that the people who get power are those who sought it. In the United States, a citizen cannot realistically hope to become President without first holding numerous lower offices, appealing to the right constituencies, raising massive amounts of money, and desperately seeking power. This does not seem like a recipe for a wise, impartial leader. Rather, it results in the nation being perpetually governed by the leader of slightly-over-half of the voting population.
Yes, a monarchy may sometimes get a bad monarch. But a republic can get a bad President or Prime Minister just as easily. And in fact, I would say that a monarch is less likely to be bad: a person who invests his heart and soul in campaigning to get power will probably be bad at least slightly more than half the time. A child born in a particular family, especially if the family was chosen for some particular virtue in its past, is probably just as likely to be good as to be bad. Thus, the monarch is not any more likely to be bad than a President, and may in fact be less likely to be a bad leader.
However, I would say that for the monarchy to work the limits to the monarch's power have to be well written. If the monarch has no power, the monarchy is pointless and superfluous. If he has too much power, he may easily become a tyrant. I would say that the monarch should be specifically prevented from doing those things which people tend to find unpleasant (torture of dissidents, massive wastes of government funds, excessive restriction on the press), allowed to do those things that have to be done by someone (approval of legislation, meeting with foreign leaders, intervening in times of national crisis), and encouraged to do those things which make the people happy and content (assisting charitable causes, advocating for the needs of the people, sponsoring celebrations). Thus, the monarch would not be useless, but would also not become a tyrant. When an evil or incompetent person becomes monarch, he will waste his time and hold back good legislation, essentially keeping the status quo. When a good and competent monarch reigns, he will move the nation in the right direction and provide good leadership to the people.