I will make a counterpoint here.
Greek society was built around Yeoman farmers who had roughly equal landholdings, roughly equal economic outputs, stood together as equals on the field of battle (the Hoplite Phalenx) and used this as justification to stand as equals in the Ekklesia as part of the democratic government of the polis. One reason the Hoplite phalanx evolved the way it did was it specifically excluded the poor (who had no weapons to effectively attack a wall of bronze) or the Aristocrats (who's main contribution was javelin armed cavalry, also an ineffective tool against a fully formed phalanx).
So whatever your innovations are, they must be first and foremost acceptable to a class of Yeoman farmers who value both their equality and their ability to hold the power of their society. Longbows or other weapons that help the poor, or stirrups and shock cavalry which would help the aristocrats, would be considered horrifying and destabilizing for the Greeks, and indeed if introduced before the Persian Wars, would probably result in Greece being embroiled in a series of civil wars between the various class elements.
As an aside, the introduction of effective naval technology which allowed the Greeks to take to sea on equal terms with the Phoenicians (and win the battle of Salamis) ended up being extremely destabilizing to the Greek Polis system. Athens granted full citizenship to the rowers because of their performance in the Persian Wars, and this large block of relatively poor, landless people allowed Athens to become "hyper democratic" compared to other city states, develop quite different interests and priorities and become prey to Demagogues. The critical naval technology? A sliding sheepskin seat pad which allowed rowers to make longer and more powerful strokes on the oars when manning a trireme.
So perhaps the best possible solution would be to introduce more effective non mechanized farming techniques, to allow the Hoplite class to expand and provide more savings and investment to the overall Greek society. The Greeks did pretty well during the Persian Wars, having a stronger agricultural base would simply provide greater defensive depth to the Greek city states, and make it much more difficult for Xerxes to advance into Greece.
The long term effect would be to make the Persians think twice about a rematch (and if the Persians didn't come back for a second round, the Greeks and eventually Macedonian King Alexander III would not be thinking about overthrowing the Persian Empire in return).