Inverting Opamp on Noisy Ground. Would an LC filter help?
This is one of those cases where an exact circuit is needed. Showing the circuits like you have make no sense because the input source is not specified. It is also unclear what the power rails are. Attaching a block called "noise source" to 0V is also unhelpful and meaningless. 

Also, 0V is 0V - that is the baseline - you cannot say 0V is noisy. However you can say the power rails are noisy relative to 0V. 

If on the other hand your input comes from something that has a 0V rail that is wiggling around relative to your amplifier's 0V then a different solution is called for. Either transformer coupling or a differential amplifier is recommended in these cases.

The common-mode choke you have in your 2nd circuit does not achieve noise rejection as far as I can see.