Most alphabets change gradually, letter by letter and generation by generation, with no single moment anyone could point to and call the change. The Greek alphabet used today is the documented exception, and the exception has a name and a year attached to it: Eucleides, archon of Athens in 403/402 BCE, the year the Athenian assembly formally decreed that the city would adopt the Ionic alphabet of Miletus in place of its own long-used local, or epichoric, script.
Archaic Athens had written Greek with a script that used a single letter, epsilon, for both the short e sound and the long e sound the Ionic alphabet marked separately with a repurposed old consonant, eta. Athenian omicron likewise did the work Ionic split between omicron for short o and a newly invented letter, omega, meaning great o, for long o. The Ionic system, already standard in the eastern Aegean cities that traded constantly with Athens, offered a more precise and increasingly familiar alternative, and the decision under Eucleides's archonship made it official rather than merely convenient.
The date is unusually well anchored because 403/402 BCE was already a marked year in Athenian political history, the year democracy was restored after the rule of the Thirty Tyrants, and sources recording that constitutional restoration record the alphabet decree alongside it. The reform did not invent new letters from nothing; eta and omega both existed already in Ionic use. What Eucleides's Athens did was choose, in one recorded act, the specific twenty-four-letter inventory that would become the Greek alphabet as every century since has known it.