ChemicalData[] error: "L-Cysteine" is not a known entity, class,

I figured it out:

ChemicalData["LCysteine", "MolarMass"]

Turns out ChemicalData["LAlanine", "MolarMass"] also works. It seems that ChemicalData["L-Alanine", "MolarMass"] with the dash is a notational exception allowed only with alanine.

One way to try to figure such things out is to use Free-Form Input.

Type a '=', type a phrase, and execute. You'll get an output cell, which you can click to turn into input cell. Then InputForm will often show you what you're looking for.

We can see that, just as @becko has already found out, that the Mathematica name for L-cysteine is "LCysteine".