What does John 19:12 mean?

Pilate clearly has no interest in Jewish religious squabbles (John 18:31). He's seen through the obvious feud driving Jerusalem's leaders to persecute Jesus (Matthew 27:18). Jesus has shown no sign of political rebellion (John 18:36–38). One would expect a governor to simply release such a prisoner and refuse to do anything else. And yet, Pilate has already had Jesus viciously scourged and mocked (John 19:1–5). He's attempted to negotiate with the crowd to release a prisoner, only to see the people choose a known murderer (John 18:39–40). This verse, with some context from other gospel writers and from history, helps explain why this is the case.

Pilate had a poor relationship with the people of Judea. His tendency to be violent, cruel, and offensive led to several incidents of civil unrest and public outrage. That was not how Rome preferred to rule conquered territory. It's likely Pilate was under a form of "last chance" warning at this point. He may have been afraid to anger the people during a major religious holiday such as Passover. Making this more difficult, Jesus' enemies have brought along a crowd (Mark 15:15) who make it clear they're ready for trouble (Matthew 27:24).

The comment recorded here is pure propaganda. Pilate already knows Jesus made no claims to earthly kingship. Those orchestrating Jesus' death (John 11:48–53) are creating a political trap. Innocent or not, the mob is demanding Jesus' death. That same mob cynically claims they are motivated by loyalty to the Roman Empire (John 19:15). Pilate will be forced to choose between two options. He can refuse to bow to the mob, release an innocent man, and risk a riot. Or he can give in to his own subjects, execute an innocent man, and maintain his position.

At this point, Pilate makes his decision, and immediately changes tactics. His main objective now will be to "save face": to deflect being humiliated over losing this political game (John 19:13–15).