The first thing to do is to clean the skin and apply a conductive gel with the electrode. The size of the electrode will be important. Too big and it will pick up noise and influence other nearby electrodes, too small and the series resistance will be higher.
These days the electrodes are disposable.
If AC coupling is used there should be little electrolytic corrosion at the electrode. The problem with AC coupling will be “de-blocking”, that is quickly getting the right voltage across the capacitor to bring the amplifier voltage into range, while still permitting low-frequency signals to pass during observation.
Back when I was in the EEG field, protective current limiters were used on all electrodes. They were once made from a resistor with an N-channel and a P-channel depletion mode FET; as a bidirectional constant current circuit. The FETs were rated to the peak voltage of the local supply.
Analog Devices have been building isolation amplifiers since then. As an example take a look at;
http://www.analog.com/en/products/landing-pages/001/7b/7b31-isolated-voltage-input.html#Introduction
Now we might isolate the entire analogue front end, powered through an isolated DC-DC converter. All the amplifiers and A-D converters would be floating and communicating back to the display through opto-isolators or an optic fibre.
You will need to select wires to the electrodes that are not microphonic.
Where the electrode wire was screened coax, the screen was not grounded but driven by a low-impedance voltage-follower from the signal so as to eliminate the cable capacitance and restore the frequency response.
The instrumentation amplifiers in the front end were once referenced to a floating average of all signals. Now we might consider looking at the difference signals wanted and convert them independently of the average. You will need to find out what electrode positions are being used these days, and what signals must be extracted from them.
Analog Devices have expertise in EEG system component design. Go to
http://www.analog.com and search 'EEG'. You will find good information like;
http://www.analog.com/media/en/technical-documentation/white-papers/ECG-EEG-EMG_FINAL.pdf
http://www.analog.com/en/applications/markets/healthcare/clinical-monitoring/eeg-measurement.html