Adafruit has complete instructions for a Larson Scanner project powered by their tiny Trinket or Gemma boards - instructions and parts list is here https://learn.adafruit.com/larson-scann ... s/overview.
My eye slot fits 17 LEDs, so that's what we built and programmed around.
A friend who's a lot who's a lot handier than me did the soldering, and added a pot so the speed of the scanner can be changed with a twist of a screwdriver.
He also modified their code to create a "comet tail." I've attached a zip file with a modified .ino file if anyone wants to take a look.
The LEDs are *very* bright, but that can be modified in the code. Planning to put a blackened lens made of milk jug plastic to darken and diffuse the light (you can see a mockup of how that will look in my profile pic).
Not the cheapest solution, but it won't break the bank

Here's the eye: