What does "soundness of judgment" mean?

Solution 1:

It comes from the following definition of sound:

–adjective
1. free from injury, damage, defect, disease, etc.; in good condition; healthy; robust: a sound heart; a sound mind.
2. financially strong, secure, or reliable: a sound business; sound investments.
3. competent, sensible, or valid: sound judgment.

Source: Dictionary.com

Soundness of judgment suggests that the course of reasoning leading to your judgments is logical and valid.

Solution 2:

Describing judgment as sound simply means it is free from errors, and can be trusted.

Thus soundness of judgment refers to the property of having judgment that is trustworthy and free from errors.