A book was published last year about this by two Protestant authors. I heard part of a radio interview about this, and they are basing themselves on the famous prophecy of St. Malachy, who supposedly had a vision of future popes and described them in cryptic Latin phrases. Pope Benedict corresponds to "the glory of the olive." (The tie-in may be that the Benedictine order has also been called the Olivetans). The next and last pope on the list is Peter the Roman.
This is where the hysteria comes in, because if he's last on the list, wouldn't that mean the end of the world is around the corner? If you search for "pope resigns" on youtube or any search engine, you will see a huge list of end-times sites that think exactly that. But don't panic or be disturbed. Keep a few things in mind:
1. The prophecy of St. Malachy may or may not be authentic, and has been the subject of much debate. But it's probably not really very reliable, even if it is fun to speculate about.
2. Even if it were true, it has been misquoted as referring to the final persecution of the Church. But the original Latin doesn't say that; it only refers to the extreme persecution of the Holy Roman Church. That has happened many times before, and will likely happen often again. For a good discussion of this point see here.
It actually reads:
"In the extreme persecution of the Holy Roman Church, there will sit Peter the Roman, who will pasture his sheep in many tribulations, and when these things are finished, the city of seven hills will be destroyed, and the terrible judge will judge his people. The End."
3. Keep in mind that some of the fundamentalists are so excited about this because they think it will spell the end of the papacy and the Roman Catholic Church, which they consider to be the "whore of Babylon." They're wrong about that.
4. There's no indication in the prophecy that Peter the Roman is going to be a bad pope. On the contrary, he is said to pasture his sheep in many tribulations. So he's going to be a good pope (if the prophecy is even true, which we don't know).
5. My concern is that all this will distort some people's attitude toward the new Pope. They will only see him through the glasses of what they have already decided must be true. But God is with the Church and he will give us the good shepherd that we need. My own view is that the next Pope will be a very holy priest and bishop. But whoever he is, he will be the vicar of Christ and our pastor and shepherd. That's enough for me!