How do I defend monarchy in an argument?
Competent absolute monarchs do pretty well, assuming you like their goals.
For example, Louis XIV was a forceful man who sort of restructured the culture of French nobility and governance so that everything centered around him and was able to leverage that into creating a period in which France gained key territories and power in Europe, as well as thrived culturally and artistically. It was a very prosperous reign.
Similarly, Catherine the Great laid the foundations for the Russia we see today; Russia at the time of her ascendancy to the throne was extremely underdeveloped and was considered backwards by the rest of Europe for a number of reasons. For example, they still had serfs in the mid 1700s. Catherine decided to drag it into the modern era, and did so by establishing boarding schools, sponsoring cultural projects, and annexing territory, all in an effort to make Russia a sophisticated and formidable nation.
Of course, this flip side of this is what happens when you get an incompetent absolute monarch, or a despotic one, or one who is competent at doing things you don't like. For example, Catherine was heavily opposed by many of the Russian nobles who liked the old ways and wanted to keep things the same, but she simply bulldozed over them. And France's decline after Louis XIV was due to many things, but one key factor was the fact that he centered everything in his government so completely on himself that when he died, he left an enormous power vacuum that wasn't filled adequately until after the French Revolution.
If you need to defend yourself against the latter argument, (from the phrasing of the question, I'm imaging a classroom debate) just remember that monarchies aren't inherently hereditary then pull an Alexander Hamilton and advocate for an elected king.
There are various forms of monarchy, and the Parliamentary Monarchy currently practiced by the Commonwealth and most of Scandinavia may be the most stable and advanced form of government we know.
Parliamentary monarchies -- in the Westminister System -- evolved in the UK from 1689 onwards. They may be the most stable form of government mankind has yet developed, and consistently yield good results.
The head of state is the Monarch, and his or her government consists of elected members of the lower house -- Parliament or the House of Commons, sometimes with involvement from the Upper House - the senate or House of Lords. The monarch decides, after elections, who gets to form a government to govern in their name.
Theoretically, all executive power rests with the monarch, usually with the assumed proviso that they will exercise it as little as possible. One practical consideration is that this leashes in the ambitious, the top job is out of reach, and essentially limited anyway. Most of the monarch's abilities are leashed in by custom and tradition -- and hence usually more flexible and prone to common-sense than law.
The head of His or Her Majesty's government is doubly accountable: Privately to the monarch, and many prime ministers have found it it useful to quietly consult with the monarch. Prime Ministers also have to answer, usually in a weekly question period, to criticism and questions from the opposition. Political partisanship, although usually inevitable, is also limited. The parties have to work together behind the scenes, in the monarch's name.
Countries with this form of government include Australia, Belgium, Canada, Denmark, the Netherlands, New Zealand, Norway, Sweden and the UK. It is no coincidence that their governments are stable, long-lasting, and by all measures of quality of governance, yield some of the world's best results.