How to select ADC resolution, considering input signal SNR and digital signal processing?
A general rule of thumb is that is you want something to not contribute to your noise budget, that it must be at least a factor of 10 higher SNR than the dominant noise source in your signal chain.  As an example, if you have a signal source that is at 300 :1 SNR, run your ADC at 3000:1 and for all intents and purposes you can ignore the ADC.

The only way to do this properly is to do a noise analysis.

Post processing (via in DSP for example) has the potential to extract out salient features from above the noise but you have to be careful.  You have to have sufficient bit depth so you don't introduce rounding/truncation errors.  You have to ensure that you are conserving the nature of the noise (gaussian/poisson pdf) or else the noise floor may rise in an unpredictable way and may not be amenable to DSP techniques.  These sorts of steps (matched filters etc.) typically at best can improve the SNR by factors of \$ \sqrt{N} \$ and often the processing cost (# of operations) often follows \$ N^2 \$ so these sorts of steps often become rapidly very expensive.  But agains a proper analysis will show this.

I would caution you against assuming that a DSP technique will automatically reduce your noise.  It is very important that you lot at your noise sources via histogram analysis to ensure that the PDF (Probability Density Function) is amenable to processing.  I.e. it appears well behaved, Gaussian or Poisson, is not multivariate and is stationary