I heard that Foucault, the French philosopher, uses this historiographic method of argumentation. What is this and is this a method of argumentation like Socrate's dialectic (not saying that they are similar)?
To explain Foucault's method of archaeology it might be best to work from an example. In the history of economic thought there are various systems of of ideas : mercantislism, physiocracy, 'classical' economics, Marxist economics, the Austrian marginalists and so on and on.
Now, one might examine any of these theories, say physiocracy (which held, to put it briefly, that land was the sole source of wealth and income in society capable of producing a 'net product' and that in a 'natural' order of society the interests of the individual and the community coincide) on a history of ideas approach, tracing how this theory gave way to another. Or one might adopt what Foucault calls a 'doxological' approach, tracing who was and who wasn't a physiocrat and whose interests the theory principally served. Or a critical approach, assessing the theory for internal coherence and predictive power.
The archaeological method is quite different. It digs to a layer beneath the theory, if the metaphor may be allowed :
Archaeology ... ignores individuals and their histories, and defines how it was possible to think in terms of either physiocratic or non-physiocratic knowledge. (D. Macey, The Lives of Michel Foucault, London: Vintage, 1994 : 163.)
In other words, it unearths, recovers, the conceptual framework necessary in order for physiocratic thought to be possible.
His archaeology is not about what was meant when e.g. it was said say that men have inalienable rights, but how (not why) the bourgeoisie's discourse on freedom emerged as a question of rights rather than say virtues.
Histories of thought are normally concerned with how things could not have been otherwise, so are a "silent murmuring" that is not apart from e.g. discussion on human rights.
Whereas Foucault's opens up a discontinuity (it's claimed) which allows us to understand history without returning to origins: discourse as it occurs, or an analysis of the study of human history, rather than mere why questions.
See the first few chapters of the Archaeology of Knowledge. I now engage in some guesswork (highlighted below) in order to illustrate the difference from a history of ideas.
In what way is Foucault not asking "why"?
The necessity of scientific laws is often said to be different from the contingency of accidental generalizations because they support counterfactual conditions, different ways things can be, what is not the case but would be if things were different.
I would guess that Foucault's "historiography" would deal with counterfactuals differently, and would ask instead, as per the other answer, in what way way for example the counterfactual of a legislature based on 'virtue' was a hidden motive to 'property rights'.
I think one can see that in his analysis of the de-instutionalization of the mad, and how this was just another way to control them.
In what way is that less "silent"?
Dialectical tensions like these play out in e.g. his analysis of historical changes to 'episteme', foundations of knowledge, one's which govern what can and cannot be thought. e.g. in modern thought, Foucault claims, "man" came to be a foundational term of our (political, social, moral) knowledge, and this meant that man must have a dual status, both constituted and constituting. This could not have existed in the classical era, without a fully fledged 'man'.
Foucault's answer to that paradox is to show that
there is never any way out of the contradiction between man as originator and man as originated
I (my consciousness) must, as Kant put it, be both an empirical object of representation and the transcendental source of representations. How is this possible? Foucault’s view is that, in the end, it isn’t—and that the impossibility (historically realized) means the collapse of the modern episteme.
So, I'd guess, likewise in his archeology.
By interrogating the historical emergence of new foundations allowing new forms of thought, Foucault is able to uncover opposing discourses, and then show how they cannot be reconciled without debunking the "episteme" they are based on.