Can I somehow leverage my willingness to work for a lower salary in job applications? In the future, I will be on the market for a temporary academic position (postdoc or visitor) in the US, in math. Because of my lifestyle and financial situation, I do not need a large salary. Ballpark, let's say I would be willing to work an academic job for about half the typical salary.

Is there any way I can use this to my advantage? I mean: is it possible to use the fact that I am willing to work for cheap to get a job that is better (with respect to some criteria like quantity and caliber of research at the university, strength of students at the university, desirability of teaching duties,...) than I would have been able to get otherwise?
There is some possibility of this, but it would work only in certain restricted ways and in certain limited contexts.  I don't know of any reputable college or university that, when considering applicants for a position that already exists, consider an applicant's willingness to accept less than the standard salary as a point in their favor.  If you think this through, probably you can see why it makes everyone look bad.

You might have some luck getting an academic institution to make a position with an unusual combination of duties and salary that is tailored to you.  Here are some possibilities:


At some institutions, faculty couples "share" positions in some fractional way: e.g. the couple together has 1.5 positions and each does 75% of the work of a full-time faculty member.  If someone had come to me with this as a new idea I would have rejected it as ridiculous, but nevertheless it actually exists in the real world.  I know several examples.
Similarly, you may be able to arrange a single position which is "part time" in some specified way: so maybe it is a visiting teaching position with half the normal teaching load and half the normal salary.  In the very short term, such positions are common.  It is not directly in the spirit of what you are asking about, but the effect could be similar.
Finally, if salary is really no object, it is much easier to get a visiting scholar position which pays no salary whatsoever.  I did this myself once: the summer after getting my PhD (at Harvard), I moved back to my hometown (Philadelphia) and got an apartment in University (of Pennsylvania) City.  I noticed that there was a wonderful library and gym a few blocks away from me, and I presented myself "cold" in the UPenn math department with a copy of my CV and my PhD.  I walked out as a Visiting Scholar.  (Nowadays I think that what I did was a little weird: why not first contact some specific faculty member in the department?  Maybe I didn't know who to contact, or maybe I was curious to see if walking in cold and presenting Ivy League credentials would actually work.  It did!  This "position" still appears on my CV.)  


If what you are interested is not covered by the above, then I would say: in general this will be hard to swing, but if you have a more specific arrangement in mind you can ask about that (e.g. here) and maybe it can be worked out.  

Let me end by saying something that I suspect you already know: if you aspire to have anything like a conventional academic career in the future, advertising yourself broadly as someone who is willing to work at a discounted rate is a pretty bad idea.  If you do seek to pursue this idea of yours, I would do so with much discretion.