AVR-Chat

>
>- Costs no more than $12 (and preferably less) to build total, including 
>populated circuit board & LED(s). In my fevered imagination, I hope that the 
>Atmel processors have pretty much all you need except for a couple of 
>transistors, resistors, and an external switch

What are you budgeting in for design, or does that have to amortize in with the $12?

You can look at Toaster oven SMD production to keep your costs low. 
I would think a panel of 10 boards would be easy to fit. 

An AVR 2343 with some sort of PN generator code, and a simple driver for the LED would work.

Power consumption will mostly depend on the LED(s) and their operating current.