I've built a project from the design here, however the voltage is far too low to light the LED. Using a multimeter, I've measured only up to 0.33 volts across the LED and only while my phone was charging on a Qi charger. How can I either increase the voltage or find a low-voltage LED?
You must build it exactly as shown in that youtube video, do not remove any wires or change any wire position or length. This circuit acts as a 2-element dipole antenna for GHz microwaves. But there is no dipole counterpoise or second wire, or "ground wire." The entire circuit itself behaves as the second antenna wire. Your long capacitor leads and different overall shape, that's a bad change, and has removed much of the second antenna-element. (Microwave RF is weird that way.)
To start fresh, copy the shape of the circuit in the Youtube video exactly. Then, to even more increase the RF coupling, connect a few-inches wire to the spot where the LED connects to the capacitor. Perhaps even hold this second wire against your phone: placed far from the phone's internal antenna (so, place the extra wire against the spot where your face usually touches the phone.)
Better: get rid of the LED and replace it with a DC panel meter: the moving-needle type. Most sensitive is a 50uA DC microamperes meter. See Jameco or All Electronics for these.
Finally, perhaps the nearby cell tower and "number of bars" will tell your phone to reduce its power. If you're in a city with good phone reception, your phone transmitter sets itself to lower milliwatts. Instead, move to a location with poor coverage, one bar, where your phone has trouble connecting. It must turn up its transmitter to maximum output.
You have an error in your circuit and or antenna.
Red LEDs have the lowest forward voltage. Deep Red and Far Red even lower.
All other colors and white are made from blue LEDs with a forward voltage of 3+ volts.
Use a super bright LED that pushes 50-100 lumens. Cree (XPE) and Lumiled (Rebel ES Color) both make very efficient LEDs.