Think about supply voltage stability and noise, namely if you are running this from some "USB" supply or especially have it connected to, your PC usb, that "5v" will
1. not be exactly 5v (USB could be significantly higher or lower and still be in spec)
2. not be stable (it can vary within it's spec)
3. probably have a fair amount of noise on it
these factors will greatly affect your measurements.
The noise can be, at least ameliorated by averaging a number of samples. The first two need an alternative solution, either a stable (and known) external analog reference, or by having the Arduino read the actual Vcc at that moment when you make each measurement - as in your other thread about the LM335.
It is unlikely that the cell is "jumping around" in voltage during the discharge (but as it warms up it's not impossible I expect depending on your load), it's more likely a measurement error from one of the above.
Think about what task you can pass to the Arduino... if you remove the multimeter from the circuit, your setup would be sufficient for you to be able to calculate the capacity of the cell (from it's start voltage to it's terminal voltage whatever you determine that as) because your Arduino can read at any given time the voltage of the cell and the voltage across the load resistor which is across that cell - exactly the same thing. Ohm's law applied with the voltage and the resistor value gives you the current, current and the time that current was observed passing for gives you the capacity. You also now have your multimeter back so can use it to monitor the voltage (and, by Ohm's law, the current too).