Strong atheism, sometimes known as positive atheism, is the belief that there is no god. Weak atheism is the lack of a belief in god. The strong position is, confusingly, the weaker position, rhetorically speaking. Declaring assertively that there definitely is no god is a bold claim, and raises questions of definitions and evidence: what exactly do you mean by god, and how do you know this thing does not exist? Declaring merely that you lack a belief is a simpler, more robust statement. A weak atheist is a judge of propositions: "Describe a god to me, and I will tell you whether I believe in it".
So I would like to propose by analogy weak libertarianism: not so much the belief that all states are bad, but merely that we have not yet seen a state that is good. I find this a very handy position - it bypasses pointless speculation over the consequences of a totally state-free society in favour of a more incremental judgement. I can state with confidence that we have too much government right now, and I wish to move society further in the direction of a smaller state, but I refuse to engage in positive claims that any particular level of government is the right level. This also avoids If-I-Were-King fantasies and Can't-Get-There-From-Here problems. Like the weak atheist, I am open to the possibility that we may reach a level of minimal statism with which I am happy to agree.
I am aware that other uses for the phrase "weak libertarianism" exist, to mean almost the same, but I hope this etymology puts it on a sounder footing.