Extraordinary Evidence & Extraordinary Claims
Too many pages have gone by for me to try to delve deeply into the conversation thus far, so I will simply dive right in with my opinion, and see how that flies.
I think most of the discussion I have seen, worrying over the word extraordinary misses the point entirely. The phrase, "extraordinary claims require extraordinary evidence", is a rhetorical device not meant to be taken literally, and concentrating on deconstructing the words is the linguistic equivalent of concentrating so hard in the trees than one fails to notice that there is a forest.
The point is that not all claims are equal, and not all evidences are equal either. I think that one average sidereal day is 23.9344696 hours long. I have significant confidence that this is correct, because the science & technology for measuring the rotation rate of the Earth are well established. So somebody comes along and says no, it's really 23.9344698 hours, based on improved measurement techniques. I am likely to read a summery report and not quibble with the results; the new number is not so far removed from the old one as to trouble my confidence (of course one can make this all more quantitative by attaching observational uncertainties, but I trust the point comes across without them). However, suppose somebody comes along and says no, it's really 22.7984785 hours, based on improved measurement techniques. OK, now I am not going to be satisfied with a summary report. I want full documentation, a complete description of what was done and how it was done, and I will want to see independent researchers duplicate the result. I will also want to see an explanation of the mistake that has been made all this time, as to how an error greater than a whole hour went unnoticed. This new number is too far removed from the old to survive my confidence unscathed. In order to persuade me to change my mind and adopt the new value for a sidereal day, one will have to present me with evidence that inspires in me sufficient confidence to override my level of confidence in the prior number.
That level of confidence is the real point of the word extraordinary. Anything in science that is held to be true has a level of confidence associated with it. Some things we think are true, but there is room for considerable discussion; our confidence level in these things is not very high. Other things we hold to be true without question; our confidence level in these things is high. Many scientists think that string theory is nearly a true representation of fundamental physics, but clearly the community confidence level is not high; there is plenty of room to question this assumption and a great deal of argument. On the other hand, general relativity is a theory held in high regard and with strong confidence throughout the scientific community. To say you think string theory is wrong is not an extraordinary claim. Saying you think general relativity is wrong is an extraordinary claim. I am unlikely to ask for much in the way of confidence inspiring evidence for the former, but I certainly will demand confidence inspiring evidence for the latter. So "extraordinary" evidence is revealed as evidence which inspires a high level of confidence, either from its form, or level of detail (or most likely both).
Confidence is not an entirely objective phenomenon, so extraordinary cannot be made an entirely objective concept either. But it works well over a community of people. General relativity & quantum mechanics, foundational concepts of modern physics, both started out as questionable propositions, but became the standard, by virtue of being able to inspire confidence through both form & detail in objective evidence. That is how extraordinary claims become ordinary claims.