LIS3DH- Understanding Z axis output
The Z axis should show about 1m/s2  when level because it is measuring the force of gravity. The noise could be vibration due to nearby fans, machinery, wind, even people walking in the area. It will pickup very subtle vibrations.

To reduce the noise, if you do not need high sensitivity, reduce the scale of the LIS3DH. Alternatively, you could handle this is in software through averaging, moving averages, etc. You could also consider implementing a software low pass filter.