Make a Joule Thief Video - www.youtube.com/watch

This is very cool! Do you think you'll be able to power other colors besides blue and white LED's? The guys in the video were saying those two colors only. I'd definitely love to hear how it is going! Thanks for posting :)

There already exists a circuit module that allows you to power a DIY LED Hoop with one AA battery, alkaline or rechargeable for $25.
prodmod.com/buy/led-hoop...tery-upgrade/
it includes the circuit and the battery holder.

I tried LED string from China before and found the wires are very thin and break very easily. They are also set at a fixed distance from eachother, so if you want to fill up the entire length of a 40 inch hoop you need more than just one set of $8 lights. I think they are only separated by 3 inches instead of 6 inches. That increases your number of LEDs and power consumption etc.
The grade of copper used in the wire is also not good which is probably why the wire breaks easily

The joule thief will be bright at first and slowly dim over time. I also don't think you can power colors like red, yellow, or orange without using a resistor. You'll need to confirm current to each LED is within the particular LED's limits.

A reliable color changing LED hoop kit that doesn't require resistors or soldering is also already available for $36 electronic parts + $15 tubing
prodmod.com/buy/led-hoop-maker-kit/

sorry couldn't resist.

or if you want to use the AA battery it would cost:
$25 AA module or $30 pre-soldered
$15 tubing
$15 Maker Kit Core (includes extra lithium battery)
total = $55 or $60

and if you don't need the extra lithium battery it would be $50 or $55 total depending on the option you choose.