The phrase "extraordinary claims require extraordinary evidence" is a common one among atheists and skeptics. While catchy, it's unclear what "extraordinary" actually means in this context. Most often it will be applied to claimed miracles and the like. Opponents will alleged this is merely a cover for the bias the person using it holds against such things. Frequently counterclaims will be made such as that accepting this standard would require throwing out many other things which can be labeled extraordinary, such as various historical events/persons or scientific findings.
What seems clear is no one can actually agree on what "extraordinary" means in a given case. For an atheist, claiming Jesus rose from the dead may be extraordinary, since it would require a god able to do things that seem impossible. For a Christian who believes such a god exists, it would not of course. The same can go for many other issues. Many people find the idea that evolution by itself could result in our existence to be extraordinary. People who accept the theory of evolution do not. Etc.
So is there some way to salvage this? Perhaps. If we take "extraordinary" to mean simply "out of the ordinary" that might work as a definition. However, again what would be "out of the ordinary" seems hopelessly subjective. For many, believers and nonbelievers alike, miracles are out of the ordinary. Yet some believe they happen very frequently. Besides, what would be "out of the ordinary" evidence? Presumably simply hearing someone say how they witnessed a miraculous event would not suffice. That would be pretty ordinary evidence. Yet if we lack "out of the ordinary" evidence (video recordings, perhaps?) most such claims will be unsupported. Naturally believers suspect this is the point of the concept to begin with.
Another tack may work. Rather than to quibble over the specifics, it might help if the skeptic pressed the believer about whether they would accept similar claims on the same evidence, and point out cases when they have not. For instance, the story of the Resurrection is told in the four Gospels. Yet they are second-hand (i.e. hearsay) and contradictory regarding who actually witnessed it. Contrasting this, the Book of Mormon claims eleven eyewitnesses who swore they saw the Golden Plates from which it was transcribed (one later somewhat recanted, but the rest did not). This would seem far better evidence by far, yet most Christians do not accept Mormonism. Perhaps then we can get into people's reasons to believe or disbelieve. A catchy slogan does not suffice.