My first concern is that the resistance through such a small piece will be too close to the limit of detection of the instrument used. The meter should have documentation that tells you the measurement error.
There may be variability due to how the equipment is used. You can minimize this by doing all measurements yourself. Even in that case I might take several measurements from one sample (for the first few samples), just to see what happens. Does it matter how the probes contact the surface of the sample?
There will be some variation due to manufacturing. You may or may not be able to detect the difference between a sample that is 10.00000003 mm and one that is 10 mm exactly.
If necessary, all samples should be cleaned in the same way.
All samples should be at the same temperature, or make temperature an independent variable in your experiment.
Ebert sir, Thank you very much for your valuable suggestion. Definitely I will follow your instructions. But my one more doubt is that can I measure the resistivity of pure copper using the micro-ohm meter? Since, copper's resistivity is ~1x10^-8 ohm-m. or should I need to go for nano-ohm meter?
Since resistance is 1.7*10^-8 ohms/m it would seem that it is also 1.7*10^-10 ohms/cm. This is 0.17 nano-ohms, or 17 pico-ohms. The sensitivity of the nano-ohm meter might not be good enough.
You could arrange 100 of these copper pieces in series to get the value into a range of the available equipment. You will then miss the piece-to-piece variability. That might be ok, depending on the research question.