Why has hypercomputation research died down? I see a lot of research on hypercomputation in the 1990's, but in more recent years there seems to be little work on the topic. Is it true that research in this area has died down? If so, what could be the reasons for it? Was this area convincingly shown to be unpromising?
It would be better if you specified what you mean exactly by hyper-computation and gave evidence for why you think it has "died down".

In any case, assuming that you are talking about computation of functions over natural numbers (and finite strings) (since I think it is clear that models for higher type computation is a very active area, e.g. CCA) and models of computation not equivalent to computability defined by Turing machines, I don't think the claim is correct, for example see CiE'05 and CiE'11. Also see the criticisms made against the claim that hyper-computation is something new:


Martin Davis, "Why there is no such discipline as hypercomputation", 2006.
Martin Davis, "The Myth of Hypercomputation", in "Alan Turing: Life and Legacy of a Great Thinker", 2004.


If you are interested, there is also some discussion on FOM mailing list starting by Timothy Chow's email about Martin Davis' article.