The criterion of embarrassment is a principle of critical analysis in which a historical account containing elements that would embarrass the author/s is considered more likely to be true, because it is unlikely you would make up something embarrassing to yourself. Even a true account is likely to omit or alter embarrassing details when possible.
Think of it this way: Let's imagine you were making up a story about an alien encounter. Would you appeal to the town drunk as a witness? Any sane person would answer "no". If you make something up, you don't want someone whose reliability can be questioned just based on who they are, you want the most credible witness you can convince to agree with you.
This makes it very significant that women were the first to discover the empty tomb. It is a well known fact that at the time and place of the resurrection, women were not very well respected. They were considered equally reliable to drunks, and an anti-Christian called Celsus actually mocked the resurrection by saying "who beheld this? A half-frantic woman". (1)
Of course, logically speaking, this is called an ad hominem fallacy. You cannot criticise either a belief or an argument based on the person who states it, because unless it's a personal truth (e.g. "I was born in 1960"), the truth will always be exactly as true whoever states it. I may not be able to say I was born in 1960, but I can say 2 + 2 = 4, and that will always be true, no matter who says so. A woman can say it, the town drunk can say it, Satan himself, the father of lies, could utter the words "two plus two is four", 2 + 2 will always be four.
But people rarely think logically. Even today, ad hominem attacks are the most common fallacy you will hear in just about any debate. Appeals to the person's age, gender, educational history, mental stability, appearance, these are all so common that it is one of the reasons I have opted to use a pseudonym for this ministry. Now, if this is the case today, imagine what it was like 2,000 years ago. As you can probably imagine, you didn't want your case to include the testimony of "a half frantic woman".
Unless, of course, that "half frantic" woman was a genuine witness to a genuine event. See, as I mentioned above, the truth value of a claim doesn't change based on the one making it. If the women who discovered Jesus' empty tomb really did discover Jesus' empty tomb, the fact that they are women doesn't change the fact that they did discover His empty tomb.
And so while it makes perfect sense to include this "embarrassing" detail if it really did happen, it makes no sense to include it if it didn't. Women simply were not seen as reliable witnesses, and so there is no situation in which they would be appealed to in support of a fabricated event. It would be far more likely for fabricators to appeal to the disciples themselves.
The criterion of embarrassment alone is enough to suggest that the empty tomb is a genuine historical fact. Jesus' body really did go missing. And so the question arises as to where did it go? The most logical explanation is that the miracle man of Galilee performed yet another miracle: His own resurrection. And if you put your faith in that fact, you can benefit from that.
References
1. Celsus, cited in Origen - Contra Celsum, Book 2 (link)