So far 65 volumes of transcripts and translations have been published by the London-based Egypt Exploration Society, which owns the collection.
The latest volume includes details of fragments showing third- and fourth-century versions of the Book of Revelations. Intriguingly, the number assigned to "the Beast" of Revelations isn't the usual 666, but 616.
My copy of Nestle-Aland (27th ed.) at Revelation 13:18 lists for the variant reading "616" only "C; Ir mss" (the 5th century uncial manuscript C and some manuscripts cited by Irenaeus). A 3rd or 4th century papyrus containing this reading would be extremely significant, although probably not enough to outweigh the many witnesses for "666."
As Bruce Metzger points out in his Textual Commentary on the Greek New Testament (1971), the numerical value of "666" is derived from the Hebrew/Aramaic values for the letters comprising the name נרון קסר, "Nero Caesar" (in Greek Νέρων Καῖσαρ) The variant "616" results from the alternate spelling נרו קסר.
For those of you unfamiliar with the numerical values of the Hebrew alphabet, the values in the above name are nun = 50, resh = 200, waw = 6, nun = 50, quph = 100, samekh = 60, resh = 200. Add 'em up!
The name "Nero Caesar" (נרון קסר) actually appears in a scroll from the Judean Desert, Murabbaat 18, dated to the "second year of Nero Caesar" (55/56 CE).
BIBLIOGRAPHY: Delbert Hillers, "Revelation 13:18 and a Scroll from Murabbaat," BASOR 170 (1963), p. 65.
UPDATE: The fragment (P.Oxy. LVI 4499) can be viewed here. Although it is pointed out that the numerology does not work with the Greek letters, the Hebrew values are not discussed.
UPDATE (5/4): I note that this post is getting, for me, a lot of hits. For laypeople, I want to emphasize these points: (1) the most likely reading of the number in Revelation 13:18 is still "666"; (2) the reading "616" has in fact been known for many years, and even the new papyrus has been known since the 1990's; (3) the variation between "666" and "616" does not materially affect the interpretation of the book of Revelation. In short, the reading "616" is mainly interesting to textual critics of the New Testament and is unlikely to make a change in anyone's Bible or beliefs.
UPDATE (5/5): See also these posts from Jim Davila, Mark Goodacre, and Stephen Carlson, each with valuable comments and links.