If one is going to list the 5 most influential (their ideas are everywhere) libertarians of the last 100-ish years, one would probably have to include at least F.A. Hayek, Milton Friedman, Ayn Rand, and Murray Rothbard.
As regulars on this blog know, for me it started with The Economist magazine near '80, and then jumped aggressively towards Rand in 1990. Since then I have been drifting back, at an increasing rate toward Hayek, who I continue to think of as the most important thinker of the 20th century.
My current position being only slightly more radical than Hayek's, I think I can articulate the moderate libertarian, or more properly, classical liberal position quite well now:
All reasonable people roughly agree on the goals of society: It is to make all the people of the society better off. Since the already-well-off need little help, we each individually, and also collectively tend to focus our bettering efforts on the disadvantaged (in skill, in money). Were we not to expect this bettering from society, we'd all be hermits. Because we tend to agree on the goals, the question becomes how best to effectuate this end.
The primary question at hand then, is how to judge which activities make a person/society better off. For instance, it is widely agreed that giving a man a fish benefits him more today, while teaching him to fish benefits him more next year. Which action is more valuable? Your metric matters. We libertarians tend to think that long-term values should tend to trump short term values. We also think that it is very important to note that my values and yours are not, and will not ever be the same. A pluralistic metric that does not pretend to impose a single value-set on all people is also important, and so we tend to prefer liberty over most other detailed values, when deciding for a society. On the other hand, economic growth advances almost all values, often even better than the direct pursuit of those values, so economic growth, with liberty, has pride of place in our pantheon of goals.
The secondary question is whether the net effect of an arbitrary government action is liable to be positive or negative. This is of course a case-by-case question. Is the net effect of Singapore's health care regime (government mandated HSAs with single-payer catastrophic insurance) positive or negative? Massively positive with 90% confidence. The question comes down to whether ALL the effects of the activity weigh out to positive or negative. An analogy may help here.
When a drug goes into a human body, it has not just the intended effect (asprin cures headaches), but also some additional effects (asprin decreases risk of heart attack, increases risk of bleeding, and can mess with the stomach IIRC). When a new law or policy goes into the body politic, it also has not just the intended effect, but also some additional effects. Unfortunately, laws passed into the body politic are passed without $700M trials supervised by the FDA, and so just as pharmaceutical companies would like to ignore the side-effects when selling new drugs, politicians can ignore the side-effects when selling new laws.
Worse, as per the broken-window fallacy from yesterday, the positive effects of laws are more naturally visible than their negative effects. On top of that, politicians should (if they're politically savvy) construct laws that show the positive effects better than would naturally be the case, and which attempt to hide the negative effects better than they would naturally be hidden. This leads us to believe that not only is it wise to check the negative effects, it is also wise to believe that all our skills of observation and analysis most likely understate the negative effects, and overstate the positives.
The nature and scope of legal side-effects is such that libertarians have acquired a presumption against new laws. It is not that laws cannot be good...just that the law-sausages that roll out of the factory are not just messy to watch being built, but often contain pieces-parts that make the whole sausage net-bad for you. While it's not 100% obvious that this presumption against new law (the bigger the law, the bigger the presumption) is appropriately universalizable, it is hard to believe that there are a lot of informed people that believe that >50% of laws passed by lawmakers have net positive benefit.
The third question facing the intelligent observer is around the trustworthiness of government actors. Should we believe that individuals in government will be any less selfish, biased, and small-spirited than any other individuals? The moderate libertarian position: Lord Acton said it first: "Power tends to corrupt, and absolute power corrupts absolutely". Governments wield more power than anyone else on the planet, to a point that (IIRC) NY City Councillors have more budget available to their discretion than Bill Gates. While Acton said it first, I think Will Wilkinson said it best: "the problem with standard public choice is that it gives too much credit to politicians by assuming they’re like everyone else and therefore it fails to capture just how exceptionally prone politicians are to narcissism, motivated cognition, self-deception, and brazen lying."
Not only do we have to worry about goals, but we have to worry about the motivations of politicians. And they tend to stink.
Finally, I personally am obsessed with the ability to fix problems once they occur. Our wimpy human brains are simply insufficiently strong to predict the future. My observation thus far is that government is almost unique in human institutions in that it almost never fixes the problems it causes by undoing things. Rather, being a rhetoric-based rather than reality-based system, it assures us that the problems really aren't it's fault and proceeds to compound them. Effective systems are the ones that especially abandon failing programs, and governments NEVER do that.
Given all this, the modern moderate libertarian sits with a strong presumption that government activity is a bad idea. Certainly we're not always right. Sometimes group activity is beneficial, and some of those times, it is even net-beneficial when enforced via threat of violence. But the presumption should be against...with the supporter of new law having a steep hill to climb in order to garner support.
I forgot to note that there is a fifth question. Question 5 is whether the government is the right social institution necessary do something. As an analogy...all parents have had children who have a nasty habit of doing something that is likely to be moderately painful, but not permanently damaging. Either I'm betraying my own anti-authoritarian proclivities or else almost everyone has tried to extinguish said nasty habit by means of persuasion, and later threats. Most parents eventually learn that allowing the child to feel the pain once or twice is a 8x more reliable method for getting the outcome (don't touch that) than is all the inducement and punishment we can dream up. How much of government activity is simply the wrong strategy. If you want people not to be poor, should we have government handouts, or should we encourage the church to provide soup kitchens and shelters, all the while being annoyingly moralistic, so as to get as many folks as can to give up their slovenly ways. My guess is that the catholic church in particular, with 2000 years of working on this problem, is in better shape to solve it than the government. I might be wrong...but the question is worth framing in any case. Should the government be the one doing this job is a question left open by all the other questions.