It seems to me that the more pertinent question is not how to convince a skeptic that CF/LENR is real but rather persuading them why they should care. That is not a nonsensical concept.
Indeed, that is not nonsensical. It is a valid question. I cannot speak for the skeptics, but what they say is: They do not believe the results are real, so they see no reason to care.
Leading skeptics such as Robert Park have said repeatedly in the mass media that all cold fusion results are incompetence, fraud or lunacy. I suppose they believe that. They are sincere. If I thought all results were fraud and lunacy, I would have no interest in cold fusion.
So, the first step is persuade them the results are real. I think most scientists who come to believe the results are real will start to care about the research. It is difficult to persuade them, because most of them will not read any papers or look at any results. Many of them claim there are no results and the effect was never replicated. Leading opponents at the DoE and Nature say that. I do not know how to persuade such people. Long ago, I decided to ignore them and concentrate on trying to reach people who are friendly to the field and interested in learning more about it. That is the audience at LENR-CANR.org.
This state of affairs is widely attributed to suppression, career destruction of people pursuing the topic, lack of funds, dying researchers, and so on. Perhaps these explanations are entirely correct and comprehensive.
These explanations boil down to academic politics. Things like this happen often in academic research. For example, during the "AI winter" of the 1990s, the academic establishment was dead set against research into neural networks. Computer neural networks were first developed in the 1950s and progress was made in the 1970, but they fell out of fashion. When I say "the establishment" opposed them, I mean that you could not publish a paper with the words "neural network" in it, and if you talked about them you would be denied tenure, or fired outright. Your reputation would be trashed and your career ended. See the book "Architects of Intelligence: The truth about AI from the people building it" for details.
Since 2000 just about all of the progress in AI has come about thanks to neural networks. Google and others depend on it for things like Google translate. It is the hottest and best approach. Yet, for 20 years, research in this technique was not allowed.
Academic politics tend to be vicious. The reaction to cold fusion is what you would expect. However, it has gone on longer in the face of much more positive evidence than most opposition does.
So the question skeptics who have no axe to grind still have to ask is: what reason is there to expect that this situation will change? According to Jed and others, we have all the verification of the phenomenon that one could reasonably demand and yet the status of the field is largely unchanged.
This situation has nothing to do with science, or logic. It is an emotional reaction. It is politics, which is behavior common to all primates. In other words, it is deep instinct, built into our very nature. It can be overcome. It has often has been overcome, as it was with neural networks. But you never can tell what it will take to overcome it. History is sometimes strange in that regard. Sometime what seems to be an inconsequential incident causes a large change in people's attitudes. Then again, sometimes this kind of opposition is gradually battered and broken down by experimental evidence. That is how science is supposed to work. But, given the situation with cold fusion, and the absolute refusal by people at the DoE or Nature to look at the data, I think that outcome is unlikely. If ever cold fusion is accepted, these people will be the last to come on board.
As an example of how long their opposition is likely to last, the Scientific American had it in for the Wright brothers. Around 1906 it attacked them as frauds and fakers, and it later claimed they did not really contribute much to aviation. When was the last attack by Sci. Am. on their reputation? In 2003, the 100th anniversary of the first flight. As Talleyrand said of the Bourbon dynasty: "They have learned nothing and forgotten nothing." As long as Sci. Am. is published, they will denigrate the Wrights, and I am sure they will also denigrate Fleischmann, Pons and all other cold fusion researchers.
So, how else can the opposition be overcome? I do not know -- nobody knows. It might be impossible. It seems likely to me we will all die and cold fusion will be forgotten. However, based on history I can suggest some ways that it might succeed --
A commercially useful, high powered cell might be revealed, and replicated. Many of the experiments have been "high powered" by the standards of the laboratory. The power has been easy to measure with confidence. It could have been measured any time in the last 240 years, as I said. But it has been of no commercial or practical value at all. Also, it has been uncontrolled, so there is no way it could be made practical. Someone might solve that problem. It seem unlikely anyone will, since there is no funding.
An experiment that is easy to replicate reliably might be revealed. This seems unlikely to happen, given the great difficulty of replicating, and the fact that experiments today must be done manually and they take months or years to do. But we cannot rule this out. If this happens, people might replicate out of curiosity, and word might get around. There is tremendous opposition to cold fusion. Talking about it will get you summarily fired in many labs. However, there are hundreds of thousands of labs in the world, and thousands of people read the papers at LENR-CANR.org. Out of all those labs, there may be 10 or 20 where people could replicate without losing their jobs. 10 or 20 might just start the ball rolling, and get others to replicate despite the opposition. Obviously, the DoE will only do it after thousands of others replicate, but we don't need them.
Someone like Bill Gates or the people at Google might start funding the research at serious levels. As soon as mainstream scientists realize there is real funding, opposition will melt away overnight. They will all put in applications for research grants. As Stan Szpak said, scientists believe whatever you pay them to believe.
In 1989, some of the most vociferous Big Name opponents of cold fusion were quietly applying to EPRI for grants to study it. I assume they wanted to cut off competition by preventing funding for younger researchers, clearing the field for themselves. This is how things are set up in the Hubble Telescope. Much of the organization is there to prevent discoveries, not to enable them. The idea is to keep young nobodies from encroaching on the Big Somebodys who run the project. See the book "Hubble Wars" for details. That's academic science for you! Corrupt to the core.
With a lot of funding, I expect the technical problems could be overcome. It seems easier than, say, building a tokamak or curing cancer. But it is impossible to know for sure.