Prydein: PRI-daen
Welsh name for the whole of the island of Britain; elsewise, the isle is divided into three parts: Llogyr--England; Cymru--Wales; and Alba--Scotland.
Geoffrey of Monmouth, borrowing from Nennius, says that the name of the island derives from Brutus son of Aneas, the Trojan, who settled the island after fleeing Troy. However, Welsh legend derives it from Prydein son of Aedd Mawr, who lead his people to the island from Ireland (IIRC). These could be of the same origin--an eponym of a leader who brought a prehistoric tribe to the island.
However, this does not explain the etymology of the name, as the above is highly unlikely. So what does "Prydein" mean?
Prydein is merely the Cymricization of the Latin Britannia, name for the island. However, it could be the other way around--there is mention of the Brittanni tribe (or Pretani), and it is thought that the name may refer to them before it refered to the island; in their language, they may have called themselves "Prydein." P often mutates to B and vice versa, same with T and D, particularly in Welsh.
Still, what does this name mean?
Among the Britons were the Picts. "Pict" is a Latin (or possibly Greek) designation, meaning "picture"--that is, the Picts are often said to tatoo themselves; the same is said by Caesar of the rest of the Britons. Now, the Irish name for the Picts is "Cruithne." It is unknown whether Cruithne means "painted" or "pictured," or if it is the name that the Picts called themselves. However, what is to be noted is that "Cruithne," if transfered from the Q-Celtic or Gaelic language to the P-Celtic or Brythonic language, would be spelled Pruddne--Prydein.
And so, by this reckoning, the name Britain ultimately referes to the Picts, the Painted People.
UPDATE: MAY 9, 2005
Upon further research, namely concerning the roots of the word for tin, I've found what may be the ultimate etymology for Prydein.
The Old Irish word for clay is crè; in Welsh this is pridd--though perhaps with the added sense of being chalky. Further back, however, the OI word credumae refers to bronze: "tin-copper." Both crè and crēd are thought to go back to the PIE zero-stem form of *kwrēt- which also gives us the root for Prydain and Cruithen, the name for the Picts, and thus the Greek's appellation of Britton. So while crēd may mean "the British metal"--tin--it may also refer to the white mark left by the cassiterite ore, the white reminiscent of chalk.
Or, in my opinion, the word Prydein may not refer to tin necessarily, but ultimately derive from a term meaning "the chalky land." Why? Chalk is plentiful in Britain, from the White Cliffs of Dover to the plains between Oxford and Bath, where the White Horse of Uffington is carved. Indeed, it is thought that the White Cliffs are what gave Britain the early name Albion--"white".
Albion, however, also refers to "the world" (in Modern Welsh elfydd). The name of the god Albiorix means "king of the world".
SOURCE FOR UPDATE:
MacBain, Alexander. An Etymological Dictionary of the Gaelic Language. Glasgow: Gairm Publications, 1982.
The Encyclopedia of Indo-European Culture. ed. J. P. Mallory. Chicago : Fitzroy Dearborn, 1997.