An Arduino with an HC-05 Bluetooth adapter took care of driving the LEDs and wireless communications, respectively. A wooden frame later, and the whole thing is ready to go.

The HC-05, a Bluetooth to serial bridge, can be found for around $5 on the internet and therefore may be the cheapest way to add Bluetooth connectivity to your project. However, its default setting… ...

The $22 kit includes an Arduino-BLE board and 6-pin header that should be fairly broadly applicable for building Arduino-based hardware that can communicate with iOS devices.