Well ideally this could be done by democratizing the Delian league as a whole, or at least enough of it for Athens to be able to agreeably absorb enough of its allies. If more Athenian allies maintained significant military contributions to the league, then a) their opinions would matter more to military calculations and b) Athens would be less able to enforce its demands on any single one of them. If Athenian leaders decide they need to win over enough allied cities, or one sends a large enough fleet to demand its admiral sits on the high command, you could see an ad hoc representative system build up for the Delian League. There's different ways this could work, but I imagine a League Senate for diplomacy and warfare. With Athens' many people, ships, and colonies, it could still dominate this body, but its allies would be represented, and slowly incorporated into a larger united polity.
Giving up any power would be hard for Athens, and wouldn't be enough to stop all rebellions either. One way to strengthen a Greater Athens is through emigration. Attica was heavily populated before the war, and OTL Athens had strong military colonies that by the end of the war were significant influences on Athenian internal policy. Heavier population transfers to Athenian allies and colonies would tie them to Athens more closely politically and physically, as well as enabling them to field larger forces themselves. With the precedent of the evacuation before the Persians, perhaps a larger civilian evacuation to allies and colonies once the Peloponnesian War started could provide the numbers for a larger and more integrated Athenian empire. Fewer civilians crowding behind the walls in Athens proper could have the double benefit of averting the great plague.
This wouldn't in itself win the war, but a stronger alliance definitely help, and Athens has plenty of opportunities to win if things stay solid at home. Mutually beneficial relationships with allies would also have positive impacts on diplomacy with neutral city states, though it might also frighten the Persians.