05:03
@olli: First, I want to be clear that the burden of proving that this claim is falsifiable isn't on me. But I will play along.
It is worth clarifying what the claim is. I see the links in the original question talking about explicitly dreaming about a particular problem... is that the same as the claim in the question?
If I am learning a new skill, and I am plateauing in progress, I will often leave it sleep on it and try again another day. Why? NOT because I think I will have a dream (that I remember), but because:
* Sleep consolidates your memories, helping "muscle memory" and speeding up some skills (like repetitive sums, etc.) [citation needed]
* Because I am tired, and when I try again I will be rested.
* Because I might be stuck in a mental rut, and coming back to it afresh means I will have a different mindset. I will likely also review where I am up to, and in doing so may spot flaws in my reasoning or technique.
If the claim is "Putting a problem/training down when you are not progressing, and picking it up after sleeping will help you reach an answer faster (in effort, if not duration)" then we can come up with a null hypothesis easily, and try to test that.
e.g. Find a crossword that typically takes an hour to solve, and a sudoku that takes an hour to solve. Compare (a) attempting one on day 1, and the other on day 2 with (b) spending 30 minutes on both on day 1, and then finishing them on day 2.
If the claim is "if you sleep, you will naturally dream about the problem you have been working on", we can give people problems and then ask what they dreamt about.
[I have seen a paper like this - they put people in a skiing simulator game for an hour or so, and then let them sleep. They woke them up and asked what they had been dreaming about. People who were only allowed to sleep for a short time reported a lot of dreams about skiing. People who were allowed to sleep longer reported dreams about related areas - falling down mountains, and the like.
[Note: I have not provided a reference to this paper; discount your belief in it appopriately.]
If the claim is "By asking yourself to dream on a certain topic, you will dream on that topic?" we could produce a similar test.
Proving that you will NEVER dream on a particular topic is hard - not least because that means you don't even dream on the topic randomly! But we can test to a clinically significant level. If you only have "useful" dreams one time in 2000, there is no practical use in trying it.