According to Wikipedia, minarchism is a term coined in 1971 by Samuel Edward Konkin III, who advocated a flavor of libertarian anarchism, and who wanted to contrast his views against those of others (the minarchists) who supported the defense against force and fraud through a government strictly limited to that role. It is an attempt to describe what Objectivists would call laissez-faire capitalism, but on incorrect premises; namely, that statism and anarchy are two opposite extremes, with a purely rights-protecting government being somewhere in between.
The Objectivist defense of limited government over anarchism does not concede that government is a "necessary evil" nor that any kind of compromise is needed to implement liberty in an imperfect world. Rather, a rational government with objective laws is part and parcel to what a free society means, and anarchism is bad in theory as well as unworkable in practice.
The etymology of the word "minarchy" or "minarchism" also invites a confusion that is already common among conservatives: that the size of government -- its budget, number of employees, number of departments, etc. -- should be a major criterion for judging whether it is moral. It is true that our statist government today is "big" as a consequence of having powers and responsibilities that it should not have, and that a proper government by Objectivist standards would, under most circumstances, be much smaller than what we have today. But the relevant principle in Objectivism is that the state is an agent of its citizens' self-defense; that is, it is limited to specific qualitative powers. It places no quantitative limits on the actions of government (e.g., measures needed in a major war of self-defense against a strong enemy) needed to achieve that end.
To avoid confusion, and to avoid making unnecessary concessions to the anarchists, it is best to describe the Objectivist political ideal as laissez-faire capitalism.