I'm going to accept Bimpelrekkie's answer because the suggestion (in the comment section) was that I read up on "differential pair circuits" and this is what got me going in the right direction. In retrospect, this question was really about how a NMOS amplifier works with weak signals and had nothing to do with a mixer. By adding "mixer" to my question I think complicated the question more than necessary.
The answer to how to amplify a weak signal appears to be "use a differential amplifier." The bottom NMOS pair (i.e., the RF+ / RF- NMOS pair) in the diagram Bimpelrekkie posted is a differential amplifier that feeds the rest of the mixer (the rest of the mixer being the "business part" as Bimpelrekkie called it).
In my quest to read up on differential circuits I came across a differential amplifier example provided in Practical Electronics for Inventors by Paul Scherz and Simon Monk, 3rd ed, p.446.
Scherz/Monk didn't provide all the parameters in their example: the signal parameters and NMOS parameters are not given. However, I drew up the Scherz/Monk example in LTSpice and used 100k resistors as Scherz/Monk did. I don't think these resistor values are exactly right for the NMOS transistor and signal levels I used in the simulation, but nevertheless it still shows amplification. Scherz/Monk go into how to determine the R values, but I didn't quite follow. (I think the bottom line is the transistors are supposed to operate near their "Q point.")
My choice of input signals: A 10mV 1kHz sine (Vsig1) and a -10mV 1kHz sine (Vsig2). 10mV is, of course, below the threshold voltage of the NMOS.
Waveforms of the two differential signals, V(vout1) and V(vout2) in LTSpice:
Differential output waveform, V(vout1,vout2) in LTSpice:
It may be hard to see, but this signal is approximately 6.3V peak to peak, from approximately -3.15V to -3.15V.