For the experienced solderer the construction of the hat shouldn't take more than a weekend; as for my friend, the novice, it will take about a week if they put in a good amount of time daily.
Step 1: Design and Parts
The hat contains 200 LEDs total, 4 shift registers, 4 buttons, 4 AAA batteries, and one Arduino mini pro.
Along with lots of wire. Making the wire longer than it needs to be is a MUST, if the wire is short and you will have to go back to fix something it make resoldering almost impossible. So make every wire long enough to extend outside of the hat, at least to the rim to make soldering possible after installing all the parts.
The height is 8 because one shift register, connected to PNP transistors, will control the 8 cathode layers of the hat. Another great number is 10 because decade counters will output 10 rows for you. For best density of LEDs 16 tall, or two shift registers, would be fantastic but require a lot more effort and resources.
When choosing the number of LEDs across it is a lot easier to make things a multiple of 8 so that they can all be handled in the same way. There is one row on my hat that must be turned on and off by the Arduino directly and it is a bother to deal with.
I assume a basic background in soldering before attempting this build. If you have never soldered before this project will be hard and painfully long.
+Mircocontroller: Arduino Pro Mini, anything could be used but I have grown to like the Pro Mini because of it's size and ease of use. Plus you can find it for cheap on ebay.
+Programmer: You'll need and FTDI cable or breakout board or similar USB programmer to program the Aruino Pro. If you choose An Arduino Uno then you will not need a programmer. If you have an Arduino it can be used as a programmer if need be but buying an FTDI cable will be much easier and faster.
+LEDs: Ebay... Search and find your choice of color and size. I used 3mm blue that are not diffused. I would recommend buying the diffused because they do a much better job not blinding you when looking directly at the hat. Though I am cheep and not diffused still look good.
+Power: I bought a 4 battery AAA holder from radio shack. I love that a switch is built in and its only $3. A 9V would work but wont last as long and need to be attached to a regulator to be used. AA provide more amp hours so if that's what you want and you don't mind the weight then pick them. You really only need 3V to run the hat so as little as 2 AA or AAA would work just fine.
+Shift Registers: I had a large batch of custom 74HC595 breakout boards made from SeeedStudio and they work great for SMD components. There is no need to do that as long as your are handy with a soldering iron DIP and some proto board would work great.
+Transistors: PN2222 are great for LED projects, buy a batch off ebay and have fun.
+Resistors: I used 402 ohm resistors because they were cheap on Newark. Anything between 200ohm and 500ohm will work. The higher the Ohms the longer the battery will last. Also you'll need some 10K ohm resisters for the buttons.
+Capacitor: Its always good to thrown in a capacitor, especially near your microconroller and important integrated circuits. I used 10pF, you can get a mixed pack at radio shack and use any of those. Capacitors help smooth out the quick variations in current that mircrocontrollers but on the system.
+Buttons: Buy them, salvage some, it doesn't matter where you get them standard dip push buttons are all you need.
+Wire: A clean and simple project is a happy one. Ribbon cable will look much nicer and be less cluttered than using individual wire.