Libertarianism is the political theory which states that the individual is the owner of his life, and that he should be allowed to seek his best interest in so far as they do not interfere with the rights of others. Underpinning this are property rights and rules governing acceptable behaviour, which distinguish the Libertarian (or more specifically the consequentialist Libertarian) from the anarchist. Libertarians accept that man must to a limited degree, be governed. The question is therefore how.
Democracy is clearly the best system yet devised, but it is prone to the illiberal baying of the mob. Checks and balances must be put in place to counteract the tendency to majoritarian tyranny. That is why there is often a revising chamber in bimarcal systems whose members have longer terms or even life tenure. In addition, many countries have constitutional courts, which check that law is compatible with the constitution.
There is no reason why the hereditary principle needs to lead to poor governance - indeed it could be argued that as people outside the democratic process, like Lords (appointed or hereditary) Kings and Queens are ideally placed to oversee lawmaking in a dispassionate way. Currently the Queen has no real political power and that is as it should be, however she retains the ability to refuse assent. I.e. she has power of veto. It is an unused nuclear option but there, should a demagogue ever gain the keys to number 10, she could refuse assent to an enabling law. Then again, she might not, but she'd do a better job than politicians, who have rarely voted for less power!
This is not rosy-eyed wishful thinking. Countries in Europe which still have Kings and Queens tend to be the ones with the longest liberal traditions of good governance: the UK, The Netherlands, Sweden, Denmark and Norway. They have old institutions precisely because they have not had revolutions. In many less stable countries, Spain for example, Monarchs have been a positive influence. King Juan Carlos intervened to introduce democracy. In times of crisis, Kings have provided better rallying points for Governments in exile than more controversial political figures - compare King Haakon VII (an elected monarch) and De Gaulle
Monarchy is an abuse of a number of libertarian principles - Clearly the Queen cannot be said to be the owner of her life, nor can her Children, especially the Prince of Wales. The Royal Family is a victim of the system of Monarchy, not its beneficiary but there is no reason to suppose that the people of the country suffer in any way for not being able to choose their symbolic head of state.
Then there is the personality of the current incumbent of the position. A better and more selfless example of service to the country cannot be found. I would rather have her as a guarantor of my rights, overseeing and guiding the lawmaking process than than an unaccountable and changeable document, or heaven forbid, the EU, which will inevitably be deeply flawed.
I am also a conservative, for the same reason I am a libertarian. I believe the state to be in most instances incompetent, therefore constitutional inaction is usually preferable. Basically, if it ain't broke, don't fix it. The Monarch does a good job as figurehead and without many of the disadvantages of more partisan political figures - especially in her ceremonial roles as head of the Armed forces and fount of honour. Any replacement of the Institution of monarchy would require a written constitution, which in the current political climate would resemble not so much the admirable document which underpins the USA, as a leftist wish-list of positive rights, which would render any future libertarian government totally impotent.
I am a Libertarian. I am a Monarchist. Are these two beliefs compatible? No, in theory, but Yes in practice. Constitutional Monarchy - better than all alternatives.