LEDs vs. Zeners: what is quiet?
(True) Zener diodes are not generally used as noise sources- 'Zeners' for >6V are actually avalanche diodes. 

This paper mentions that a 5V Zener has about 15-20dB less noise than a 12V 'Zener'. So, I'll assume that the (dominant in a 12V 'Zener') avalanche noise is not a factor since you are going to use a true Zener diode. 

Both an LED and a Zener will have shot noise, which is proportional to the square root of current, so it may well depend on the relative operating currents. There will also be some thermal noise from the resistive component, which would probably favor a Zener over a small LED.

Usually if noise is concern you should follow a relatively low noise source with a conventional low pass filter or something like a capacitance multiplier. That's how reference chips designed for RF circuits achieve uV wideband noise using relatively noisy bandgap references.