In my days as a YouTube apologist, I heard this claim a lot. "Extraordinary claims require extraordinary evidence". But this statement is loaded to the brim! Before we explain why, we need to know what is meant by the word "extraordinary". A standard definition would be "very unusual or remarkable".
With that in mind, we can say that the claim itself is extraordinary. It is, in fact, unique. In what other discussion is the standard of evidence raised so high? In most conversations, the facts are presented, and those facts are used to draw a logical conclusion. It really is that simple. But when it comes to Christianity, the ordinary evidence is quite compelling, and so atheists are forced to switch from "there is no evidence", a statement which is easily disproven by the presentation of that evidence, to "that evidence is not extraordinary enough". There is no other discussion in which this would be considered logical, and thus the claim itself is extraordinary. But there's no evidence to back up this extraordinary claim, and thus it is invalid by its own standards.
But as I said, the statement is also loaded, as it assumes there are any extraordinary claims in Christianity. But in what way is Christianity extraordinary? For 6,000 years, human beings have known God, at least in part. For 3,400 years, God has been revealing Himself in written form. For nearly 2,000 years, those revelations have been complete. For 500 years, those revelations have been widely distributed in a number of different languages. In other words, Christianity has existed, in whole or in part, and has been believed, in whole or in part, for the entire history of the human race, and by a significant portion thereof.
Aside from the true God, there has always been a concept of the divine in human beings. Indeed, we are literally born with it. Even the majority of atheists have a sense of Godliness, which they display every time they talk about morality. In other words, there are very few "extraordinary" claims in theology.
By contrast, atheism really is an extraordinary claim. It's an old one, even being visible in the Bible (e.g. Psalm 14:1), but for the majority of history, it has been ridiculously rare. As Isaac Newton once said, "Atheism is so senseless and odious to mankind that it never had many professors." (1)
In the modern day, atheism is still a minority position, but is nevertheless growing. This, unfortunately, is due to the formulation of a new religion called Evolution. Evolution, as defined by Gerald Kerkut, is "the theory that all the living forms in the world have arisen from a single source which itself came from an inorganic form." (2) Evolution also encompasses the planet, the solar system, and even the universe itself. Effectively, it is the atheist's creation myth.
Having studied, at least in passing, a number of the world's most popular religions, I can honestly say there is nothing more extraordinary than Evolution. While there are some shockingly unbelievable claims in other religions, Evolution actually claims to be the backbone of science while simultaneously ripping science to shreds. Several well established scientific laws, where they would normally be respected almost without question, are thrown out of the window in defence of Evolution. Where evidence is usually used to judge a theory, Evolution is used to judge the evidence.
Thus, if extraordinary claims require extraordinary evidence, you're far better off just becoming a Christian. No evidence could be extraordinary enough to justify Evolution, or atheism in general, and no claim on Earth is any less extraordinary than that there is a God who created us, and loves us enough to seek a relationship with us.
References
1. Newton, Isaac, cited in Memoirs of the Life, Writings and Discoveries of Sir Isaac Newton by Sir David Brewster, 1855
2. Kerkut, Gerald A. - Implications of Evolution, Pergamon, Oxford, UK, p. 157, 1960 (link)