What does John 10:34 mean?

Prior to this, Jesus was about to be stoned by a mob of hostile religious leaders. Their claim was that Jesus committed blasphemy, by claiming He and God were "one" (John 10:30–31; 10:33). Jesus' immediate response, however, was to refer to His miracles—indirectly pointing out that His words ought to be judged in the context of His ministry, and not with careless prejudice (John 10:32).

Starting here, Jesus begins to use a debate style which these men knew all too well. Religious leaders and scribes of this era would often debate Scripture, using an endless barrage of technicalities and convoluted explanations. Jesus engages in an abbreviated form of that technique to prove that, even by their own standards, they're being hypocrites.

Jesus cites Psalm 82:1–8. A reference to the Old Testament grounds His claim in something these men claim to take seriously: the Word of God. In the next verses, Jesus will compare the words of the Old Testament to the claim these men now claim is blasphemous. It's important to note that Jesus isn't making a blanket defense of all claims related to God. Rather, He'll once again point to all of the ways in which He fulfills the role of Messiah. If "gods" is an appropriate description—from God Himself—to those who are enabled to perform His will, then how can they stone Jesus, who has performed actual miracles, for blasphemy by claiming to be the Son of God?