“Aeacus while he reigned in Aegina was renowned in all Greece for his justice and piety, and was frequently called upon to settle disputes not only among men, but even among the gods themselves.” A Dictionary of Greek and Roman biography and mythology William Smith, Ed.
The first of the dictionary’s references agrees that Aeacus was called upon to settle disputes among the gods; (Aegina) “bore to the lord of the loud thunder the best of men on earth, brilliant Aeacus. He was judge among the divinities even,” (Pindar. Isth. 8.48)
But in all of Greek myth I can find no specific details.
The second and last reference the Dictionary uses for the opening quote is;
“Sciron (King of the Megarians) married the daughter of Pandion and afterwards disputed with Nisus, the son of Pandion, about the throne, the dispute being settled by Aeacus, who gave the kingship to Nisus and his descendants, and to Sciron the leadership in war.” Paus. 1.39.6
This might be a fine example of Aeacus’ skill as a mediator among mortals, but that’s not the gods. It does remind me of Briareus, the Hundred-Hander when he mediated the claims of Helios and Poseidon for Corinth;
"The Corinthians say that Poseidon had a dispute with Helios about the land, and that Briareus arbitrated between them, assigning to Poseidon the Isthmus and the parts adjoining, and giving to Helios the height above the city (the Acropolis of Corinth ." Pausanias, Description of Greece 2.1.6
The nice thing about Briareus’ judgement is it gives to each god what most rightly would be associated with him; the sea-god Poseidon gets the watery channel and low lying land and the Titan of the Sun gets that portion of the community closet to his realm in the sky, the acropolis. The truly marvelous thing about Briareus wise judgement among the gods is that afterwards; no one’s wings got torn off (Paus. 9.34.3), their river beds didn’t dry up (Apollodorus, Bibliotheca 2. 13), they weren’t skinned alive (Apollodorus, Bibliotheca 1. 24) nor had their plains flooded (Apollodorus, Bibliotheca 3.14.1) Assuming, Aeacus made similar Solomon-like judgements among the gods, what were they? I propose two disputes we know he was involved in. Maybe his involvement hints at the notion he was the mediator in each case;
1. "When for marriage with Thetis there arose strife 'twixt Zeus and glorious Poseidon when each of the two gods would have her to be his lovely bride.” (Pindar, Isthmian Ode 8.3) No admittedly Pindar gives Themis all the credit for judging between the two and picking a mortal as the nymph’s husband. And according to Aeschylus, maybe it was Prometheus that made them see the light. (Prometheus Bound) But still the choice of Aeacus’ son Peleus as the spouse of the Nereid, must if not indicate involvement at least indicate close association with all involved.
2. The revolt of Poseidon and Apollo, against Zeus. The sole reference to this revolt is in Iliad 7; Poseidon speaking to Apollo “the woes that we twain alone of all the gods endured at Ilios, what time we came  at the bidding of Zeus and served the lordly Laomedon for a year's space at a fixed wage, and he was our taskmaster and laid on us his commands.” So who mediated this solution to their strife? No reference to Themis intervening. Could it have been Aeacus again? “Aeacus, whom wide-ruling Poseidon and the son of Leto, when they were about to build the crown of walls to encircle Ilium, summoned as a fellow worker” (Pindar Olympian 8.34-35)
All this is pure speculations on my part! But, if some ancient forgot legends say Aeacus settled disputes among the gods I judge two of the disputes were over;
· the hand of “a girl, just one girl” and
· a revolt against Zeus
Anyone got other suggestions?