How did the notion of political veto come to be, and by what rights or criteria did the nation/state/individual come to such power?
The romans created it. Veto means "I forbade" in latin and the right of veto was the privilege of the tribunes of the plebs. When one of them stood up to say "Veto" it blocked the decision. Ironically what was originally a tool to protect the plebeian against patrician abuses of the beginnings of the republic became at the end of it one of the base of the imperial power as Augustus received power over the "imperial provinces" and the tibunita potestas, the power of the tribune (of the plebs) enabling him, the patrician to veto any law he would not like.