Measuring small movements of a needle and syringe
An accelerometer is definitely out in terms of noise.

A mechanical arm system, while potentially accurate enough, may well influence the injection scenario enough to render your results meaningless. I suspect a student struggling to control the position of a small syringe would be distracted by a large measurement arm, no matter how well balanced and low friction.

The only real \$^*\$ options you have are optical.

It should be possible to mark the syringe at both ends of the barrel with fiducial markers. The resolution you can achieve is limited by the optics for pointing multiple cameras at the target. If the test site is small and the location well defined, then you can use zoom optics to make the image fill a significant amount of the frame. HD cameras, and sub-pixel location of the fiducials via something like OpenCV ought to make your target resolution achievable.

\$^*\$ real => low cost, keeps the imaging volume clear, and it's apparent how you'd get the resolution. There are plenty of other modalities, for instance MRI, PET tomography, ultrasound, magnetic tomography, Xray CT, resistive tomography - which need variously calibration, development, expensive equipment etc.