Well, in every arcade versiom his name is spelt 'Raiden'. It has been confirmed that is the official spelling by Midway, but was changed to 'Rayden' in the MK1 + 2 home versions to avoid a copyright infringement. Another game (or a chinese film - I've heard both) used the name 'Raiden' and they didnt want any legal trouble.
I think they finaly reverted back to the 'Raiden' spelling for the MK3 home versions, in the intro it says 'these are the words of Raiden'. In MK4, DA and D it is also spelt the classic arcade / proper way.

So there you have it...

I actually believe it was after MK3 that they changed it back to "Raiden." I say this because on MKT for the PSX it is spelled "Rayden."

raiden's name was rayden in all mk3 home versions except for mat2. which is an emulation. not a home version. i have them, i know. sega snes psx , even umk3 saturn.

I guess it's all cleared up because it wasn't a problem for Kinami to use "Raiden" in Metal Gear Solid 2.

My god given name is Raiden, but as my fellow internet colleague mentioned because of the whole copyright crap I changed it for a little while. But now I'm back and ready to kick ass and take names.

Interesting choice of words, there, Thunderg0d: ''god-given name".

In Japanese mythology, the god of thunder was named Raijin (Rai - thunder or lightning, Jin - god or deity). Americanized, the name became Raiden, pronounced Lri (long 'i', 'l' and 'r' sounds smushed together)-Den. That proved to be harder for Americans to pronounce, so instead, they changed it to Rayden, prounounced RAY-den.

Raiden in MK was always intended to emulate the Japanese thunder god he was based off of, but for a while, from what I've heard, it was thought that the name Raiden was copywrited (see Raiden from MGS, etc.), so they changed the name to avoid legal implications, as well as make the name easier for fans to pronounce. Later, they apparenly learned that the name COULDN'T be copywrited (since it belonged to a Japanese thunder god and not any particular company), and the name was changed back.