How to measure voltage drop along silver foil with nanovolt precision, using Arduino's ADC?
The Arduino has a 10 bit ADC with 1024 levels, so the minimum change you can detect with your method is 1 part in 1024, ie. ~0.1%. In practice you need to keep within the ADC's input voltage range and above its noise floor, so the practical sensitivity will be lower (perhaps 0.2~0.5%).  

However that's not your biggest problem. To reliably detect small changes you need very stable analog circuitry that is not affected by confounding factors such as wiring resistance, thermocouple effects and amplifier drift. 

You are using a 4&ohm; resistor to set the current to a fixed value, presumably by connecting it to +5V. This will draw a bit over 1A so the resistor will have to dissipate more than 5W, and it will get hot. As a result its resistance will change, and so will the current. Supply voltage variations and changes in circuit resistance will also affect the current. 

A more constant current can be created using a 3 terminal voltage regulator (eg. LM317) in current regulator mode. The regulator should be attached to a large heatsink to keep it cool.

When dissimilar metals are joined together and exposed to a temperature difference, a small voltage may be produced (Seebeck effect). To minimize this you should try to keep the wiring and connections at a constant temperature (keep that hot resistor well away from the rest of the circuit!). 

Joints may have randomly varying resistance which could introduce noise into your voltage measurement. Where practicable you should solder all joints, and use tightly screwed or bolted connections for the rest (no breadboards!). You should make a Kelvin (4 wire) connection to your silver foil, so that small resistance variations in the end connections don't don't affect your readings.            

DC amplifiers drift and produce noise, both of which are amplified along with the signal. You should use a 'zero-drift' or "chopper stabilized" op amp, and take the average of many readings to reduce noise.   

If you need greater sensitivity then you can make a differential measurement between the foil and a reference which is adjusted to the same initial voltage. This is commonly called a Wheatstone Bridge. You can then increase amplifier gain without the signal becoming too large for the Arduino's ADC.