We have precious little access to how Euthyphro conceived of wisdom because Socrates so quickly set the terms of the debate to suit his own preferences. However, the dialogue does provide us with glimpses of how Euthyphro and his ilk conceive of wisdom, even if only through the distorted lens of Plato’s portrayal. For to properly demean his potential rivals, Plato must at least provide his readers with a believable image (one might say ‘simulacrum’) of their methods.
Socrates distinguishes two reasons why he is being brought to trial. He presumes to “teach his own wisdom” and “makes others to be like himself” (3c-d). His wisdom, as we know, is the wisdom of ignorance and doubt, the wisdom of not knowing. In making others like himself, then, he leads others to doubt and ignorance.
By contrast, we can probably assume that Euthyphro did not attempt to make others like him. For one, his self-esteem seems to rely on his conception of himself as different from the population, of having a higher source of knowledge and a deeper insight into divine matters than the majority.
More to the point, though, he understands the dangers of taking a position like Socrates directly against the values of the polis. The mockery he endures and the divine origin of his claims makes him appear much less threatening than Socrates. When Socrates describes his direct approach, Euthyphro voices that he has “no desire to test their [the court/polis] feelings towards me” as a bearer of personal wisdom.
Notice, Euthyphro doesn’t argue about having personal wisdom, just about whether he would be willing to pit his personal wisdom so directly against the values of those with authority in the polis. He implicitly accepts that he has his own wisdom, his own daemon. Where he differs with Socrates is his manner of expressing that wisdom.
Superficially, we might argue that Euthyphro lacks the strength of his convictions, lacks the will to take a stand against false values. And yet, Euthyphro is prosecuting his father for the death of a murderous servant. Euthyphro places his own values contra the polis as Socrates does, except that unlike Socrates he has concrete values to offer in their place.