There was no name for the group, at least not as preserved in the sources. "Delian League" is a modern terminology based on the fact that Delos was the de facto headquarters of the alliance. Thucydides (1.97) essentially calls it a hegemonia and distinguishes that term from what the group would become later -- an arche (what we usually term "Empire"). In the beginning it was a collection of willing and autonomous allies with a common goal: the continued fight against the Persians. The Spartan Pausanias had been slated to be leader, but his overbearing behavior got him ousted and the Athenians were asked to "lead" instead. We don't know exactly what this means, and though we know that the group met in a congress, we don't how exactly decisions were made in the early period. Eventually, of course, the Athenians would come to call all the shots, though we need not but a lot of stock in [Aristotle](Ath. Pol. 12.2.2), which accuses the Athenians of tyrannical behavior from the very beginning. No source mentions any symbol or the like for the group.

A great place to read about the Athenian Empire is:

Morris, I. 2009. "The Greater Athenian State," in The Dynamics of Ancient Empires, I. Morris and W. Scheidel, eds., 99-177.