How to characterize SNR for an amplifier with extreme low input signal?
Your amplifier has average noise characteristics, the problem is that your signal is very, very weak. The amplifier is responsible only of the noise part of the SNR, so it has not a "poor SNR", but a "poor input referred noise".

To obtain a better SNR you can either amplify your signal, without adding noise, or reduce the noise.

Since amplifying without adding noise is quite a task I'd say that searching for low noise op amp is what you should do. A quick googling landed me right on the TI low noise amp page. As you can see there are sub \$\frac{nV}{Hz}\$ amplifiers, which may be a good start.

Please note that if your signal is low pass with a narrow band (such as and EEG) you need to consider also flicker noise and offset. If this is an issue you should really switch to an instrumentation amplifier that includes a chopping modulattor or at least an auto-zero or a correlated double sampling input stage. This first amp can help you give a boost to your signal, somewhere around 40dB, you can then add another stage to add the rest of the gain. Please note that in a multi stage amplifier the first stage is the most important noise wise, i.e. your second stage may be much more noisy than the first.