You could design a voltage divider and use the wire itself as one of the resistors. This would give you a voltage drop, which could be measured on your analog pin. You would want to design the resistor value to give you a good range of voltage drop. What range are you expecting to see in resistance of the wire? The smaller your range, the more accurate you can get a measurement.
D2 -- resistor -- A
A2 -- A
D3 -- resistor -- B
A3 -- B
D4 -- resistor -- C
A4 -- C
AB
D3, A4, and D4 set to input without pullup
Change D2 to an output and set High
Change A3 to an output and set Low
Change A2 to an analog input and do an analogRead
The resistor is selected be low enough to put the voltage across the phase within a useable range, but high enough not to overload the output of the Arduino (absolute max 40mA, better to stay at or below 30mA max).
If you use the internal reference voltage, low voltages may not require amplification.