Powering an LED from RF (cellphone signals)
Despite the contrary opinion from my most learned colleague, I suspect that you may be able to achieve a usable result.

The units shown used tuned antennas which are resonant at the cellphone frequency.  

The total length of wire required is ~~~  L = 150 / F
 L = length in metres of a HALF wavelength.
 F = Frequency in MHz.

If you stare into the beam (with remaining good eye) a modern LED will produce visible light at 1 mA (and less), reasonably bright output at 5 mA and very bright output at 20 mA.
 1 mA = ~~ 3 mW
 5 mA ~~~= 15 mW
 20 mA ~+ 60 mW.

To get even 3 mW range will need to be small. 

The video footage MAY be faked (it being on You tube) but seems more liable to be real.

Schottky "hot carrier diodes" should work OK.