Does this mean anything other than something becoming an issue or a question? And what does that mean?
It's probably true, to start with, that French academics do perhaps use the word "problem" where an English academic would use either "issue" or "question", and in a way that is more or less synonymous with the latter. It is certainly the case that where an anglophone academic might refer to their "topic", subject area etc, French academics will commonly refer instead to their "problematic". On that level, we can just see it as a stylistic element without deeper significance.
There is also however a history of the notion of the "problem" in philosophy that I did some work on a while ago—based especially on an article by Emile Bréhier, "La notion du problème en philosophie", which actually started out by saying that we bandy about this word in a way that seems more or less synonymous with other words, like "question", and asked whether it had a more specific sense.
There are some aspects of the term that may be relevant to Foucault's use, though I wouldn't want to claim he is consciously referring to these in his choice of words.
The earliest technical sense of the term "problem" was in geometry, where it can be defined in contrast with the "theorem".
- A theorem demonstrates properties of a figure (eg. the triangle in Pythagoras' theorem) that can be deduced from the definition of that figure.
- A "problem" also indicates properties of a figure, but these are not ones that can be deduced from its definition, but can only be shown by carrying out an operation on the figure.
This difference actually gave rise to philosophical debate (cf. the commentary on Euclid's Elements by the Neoplatonic philosopher Proclus) regarding the relative status of problems and theorems in geometry, as problems seemed to defy the image of truth as eternal and belonging to the intelligible realm alone by suggesting that there were some truths that were actually generated in an operation taking place in time and space rather than being contained in their intelligible essence.
But, moving on from its origins, a "problem" is still, in everyday parlance, a phenomenon which demands both thought and some kind of practical intervention, these often being fused together: if there is a "problem" with my shower head, I have to do something to determine what the problem is and resolve it, and these actions are also stages of development of my knowledge. A thinker like John Dewey is someone who places the notion of the "problem" at the center of his epistemology because of this way it combines intellectual and practical elements (or rather shows how they can't be separated) and also shows knowledge to have stages rather than simply being an abstract relation between subject and object.
Bringing it back to Foucault, if we consider the way Foucault refers to something, eg. sex, being "problematised" at a given era, it is clear that for Foucault this means both that becomes an object of theoretical investigation, something to be known, and also an area of practical intervention: in that way the notion of a "problem" captures something of the power/knowledge fusion better than if we said it becomes an "object" or a "question".
More generally, if things—eg. the subject, truth—are "problems" in Foucault, this implies, from a geometrical perspective, that these are terms whose sense is not given in their definition but relies rather on a process of construction.
I'm not saying that Foucault is implicitly referring to geometry, but I do think the choice of the term "problem" (of the subject, truth, etc.) rather than, say, "concept" or "theory", does bely a difference of this sort, a difference which certainly is carried on in certain philosophical debates (Kant, for example, who made his name arguing against the worth of "analytic" truths, ie those deducible from concepts, and suggested all concepts, even geometrical ones, must be constructed—his term—in experience).
