At a little over $1/each (less in bulk) the ATTiny85 is a great chip for the job: it has 6 I/O pins and is capable of 16Mhz with minimal external components, is enough to work with many sensors, even drive popular addressable LEDs like Adafruit's NeoPixels, WS2811 strips.
The setup process is a little technical, but not too scary I promise, and you can use the familiar Arduino programming environment.
What you'll need:
* An ATTiny (this tutorial uses the ATTiny85-20PU)
* a USB ISP AVR Programmer (My tutorial uses this one)
* a breadboard
* some jumpers or wires (color coded wires are really helpful)
* a computer with the latest Arduino software installed (if you don't already have it, download here)
* a 5V wall wart (Old cell phone power suppliers work wonderfully, double check them for a 5V output. If you need one, these are plentiful at thrift shops!)
* 5V Addressable LEDs (this tutorial assumes use of 3-wire WS2811/WS2812)
* 10 uF capacitor
* 1 low value resistor (ex 47, 100, 220 Ohm)
* A plain 1-color LED for testing
If you're looking for addressable RGB LEDs try Adafruit's Neopixel line or WS2811. They come in strips, pixels, segments, modules of many varieties. Working with 5V (as opposed to 12V) will be easier for this tutorial, so that's what I recommend!