I have written about this question in my blog. I appreciate your opinions.
Digitally speaking, the SNR should remain the same when you simply increase level. The SNR in the digital world is the same as the SQNR (Signal to Quantization Noise Ratio), which is determined by your bit depth. 16-bits gives you 96 dB SQNR (ideal case, assuming your signal is captured at maximum level before clipping), so if you captured some quiet audio at say around -30 dBFS, and normalized the audio file to 0 dBFS, your digital noise floor would also increase by 30 to -66 dBFS. That's why with quiet or unpredictable recordings, using 24-bit gives you more leeway to boost levels in post without worrying about your quantization noise. In the previous example with 24-bit your SQNR would still be less than -100 dBFS after normalization.
Of course if your audio signal has captured a higher level of analog noise (i.e. background ambient noise or circuitry noise), that will take precedence over the quantization noise, but once digital should still increase linearly as you boost amplitude. Hence as @TheFaderJockey says get your analog signal hottest at the preamp gain to minimize noise ahead of digitization.
In the analogue world, the "gain knob" at the top of your console is often actually an attenuator, rather than an amplifier. The mic pre boosts the signal by a fixed level, and the gain knob actually reduces that level as the knob is moved to the left. Add to that the fact that the preamp is probably the cleanest sounding amplifier circuit in your signal chain: introducing the least amount of noise.
Therefore, for the highest SNR, the best practice is to get your gain up as high as you can right there at the pre. Set it so that you have just a little headroom before clipping, and you'll have the highest SNR you can achieve.