meaning - What does "soundness of judgment" mean?

Soundness means stability, validity, rational, free from error, reliable and so on. Lets see it in different context first.

"During the past few years we have questioned the soundness of many principles that had for a long time been taken for granted". Here soundness relates to the validity or rationality of these principles.

Another example can be "The politicians talk about the soundness of the instincts of the people. Something more than an instinct is wanted in a democracy." Here the focus is on the reliability of the instincts.

Another one taken from a newspaper excerpt, "One has to question whether he really has a soundness of moral character and honesty to lead effectively." The soundness of moral character refers to substantial and enduring character.

Now coming back to judgement, it's defined as

the ability to judge, make a decision, or form an opinion objectively,
authoritatively, and wisely, especially in matters affecting action;

So, when we use soundness of judgement, we are again looking at the stability, reliability or the ability to look at the opinion as valid, rational, and strong enough to stand by it.

As in, "He was a man of sound judgement". You can get from this sentence that the person has a good sense of discretion and his decisions/opinions are in fact sound.

Some more examples just so you can relate to them-

"If this is true, if past experience can be used in present judgement whether it can be recalled or not, we should expect ones power to recall and apply it explicitly to have very little to do with one's soundness of judgement. "
~The Nature of Thought

"The king, by giving ear to these rumours, has perverted the soundness of his judgement"
~The Vagaries of the Qasidah by J. E. Montgomery