Cheapest way to power an LED from an Android phone using Bluetooth
Bluetooth is a complex protocol. Handling it really requires some kind of programmable state machine – in the end, a processor. So, even the least complex bluetooth device will always contain a dedicated processor, for the sole purpose of &quot;talking&quot; bluetooth.
It also needs to contain a lot of radio hardware – DACs and ADCs, oscillators, amplifiers, mixers…, which ideally should be placed on a single IC, to avoid complexity and imperfection.
Therefore, standards like Bluetooth, which are meant to go into a lot of cheap consumer devices, are designed with one thing in mind: Integration into ICs that also contain other functionality.
Therefore, literally hundreds of Systems on Chip (SoC) containing both a bluetooth frontend and a programmable microcontroller exist. A module with such an SoC sounds like what you need (example). Most of these are cheaper than official Arduino boards.
By the way, don't underestimate the complexity of bluetooth: Compared to what a bluetooth frontend needs to do to receive a couple of bits, whatever you've been writing on an Arduino will look very small. Luckily, the SoCs mentioned above do all this for you. You just have to write a very simple firmware for the microcontroller on the same chip as the bluetooth controller.