What would be the best, cheapest, smallest way to power a Bluetooth bone conduction headset project?
I won't promise this is the &quot;best, cheapest, smallest&quot; way to power your bone conduction hat, but I recommend you consider zinc-air cells, better known as a &quot;hearing aid battery&quot;. Why:

Cell voltage is 1.4 V, just connect 4 in series to get 5.6 V.
High current capability
Disposable, instead of rechargeable -- easier for beginners
Environmentally sound -- no dangerous elements, toss in trash
Lightweight -- one of the two chemicals is just air!
Safe next to your head -- used in thousands of hearing aids
PCB holders easily available
Inexpensive when bought in reasonable quantity

Zinc-air cells are different from other technologies in that they have a fixed lifespan of about 2 weeks. When purchased they have a seal that keeps air out. Once you pull that off, the chemical reaction begins whether you draw current or not. Look for ones designated for cochlear implants for the highest current capacity.