On the experimental argument, I largely agree with THH. It is a shortcoming that must be addressed that there is no "lab rat" experiment that can be easily replicated, with results well above the noise floor. Hopes have occasionally been raised over the years that such an experiment might have been found, but such hopes have not yet come to fruition.
On the theoretical side, I am largely persuaded that unexamined assumptions and misinterpretations of what expetiments there are abound, both among CF'ers and among mainstream physicists reviewing the body of work from afar, and that through polemics these bad assumptions have been partly codified into a CF dogma, a situation that has helped to lead theory building astray. It does not matter what an intelligent mind such as Peter Hagelstein comes up with if important starting assumptions are incorrect or are experimentally untested. If CF is eventually accepted into mainstream science, I assume that all that will be needed will be some minor adjustments to and elaboration of existing physics and one or two happy coincidences in areas of experimental physics that were up to now poorly explored. There are surely tantalizing possibilities that students of nuclear physics will have stumbled upon in their studies generation after generation, only to be told by their professors with a little too much confidence that while the idea is an interesting one the experiments do not support it. It is in such possibilities that I find the most promising avenues for an explanation for LENR.
The possibility that radioactive decay rates might be modulated is one such possibility. The idea will have occurred to any physics student after learning about the Gamow theory. The fact that the possibility is raised as a question from time to time on physics.stackexchange.com is an indication that the suggestion is not a radical departure from existing physics. There is even a page at MIT that documents a number of (non-CF) studies that have been done on variability in decay rates. If radioactive decay could be accelerated, many CF experimental findings could be explained. There may be one or two other such possibilities.