As far as hardware goes, LEDs turn out to to easily support 10-30 flashes/sec. https://www.sparkfun.com/products/531 is $0.95, more than bright enough, and will work fine directly wired to a 3.3v arduino. I've ordered two (one for each eye) to be wired into an old sleep mask :p
If you have a ping-pong ball, you can make some ganzfeld glasses as an alternative. If you want to know how to achieve this effect naturally, either from a sleep or an awake state, you should checkout forums related to AP, OBE, and Robert Monroe.
That said, building a $8 DIY version sounds like fun. Does anyone have more details on how the light is supposed to work? :)
edit:
https://www.dmt-nexus.me/forum/default.aspx?g=posts&t=30153 has some good discussion of this and similar projects. Specifically https://en.wikipedia.org/wiki/Dreamachine, which is fully specified and was apparently pretty popular back in 2005: http://www.nytimes.com/2005/01/20/garden/20mach.html
As far as hardware goes, LEDs turn out to to easily support 10-30 flashes/sec. https://www.sparkfun.com/products/531 is $0.95, more than bright enough, and will work fine directly wired to a 3.3v arduino. I've ordered two (one for each eye) to be wired into an old sleep mask :p