Arduino on battery and Conductive Thin Thread

I want to control 3 LEDs (turning "ON" only one LED at a time) with arduino powered by a battery.
What is the best solution considering that standard arduino consumes more than 15mA,
just sitting there doing nothing and killing battery consumption?
I need a starting point, which model to use (Standard Arduino, Arduino Pro Mini, Arduino Lilypad...).
Can I connect standard arduino and LEDs with Conductive Thin Thread - CTT (LEDs would be hanging on thin ropes
which would be coiled around the CTT)?

I have found this link but it's to complex to build!

What's left to me? Arduino Pro Mini, Arduino Lilypad??