Really, Zeus' lightning bolts should probably be considered more of a trademark or a calling card than a symbol. Nike has its Swoosh, McDonalds sports the Golden Arches, Batman is never without his trustee Batarang, and Zeus carries around a fistful of raging thunder. You're never going to find one without the other. That said, given that this is the first time that Zeus ever hurls an angry energy bolt we're forced to take a step back and consider what the Cyclopes' gift to the Olympian means on a larger scale.
Quick recap: Zeus releases the Cyclopes and the Hecatonchires from their prison in the bowels of the earth after Gaia tells him that doing so will win him the war. We've already mentioned that Gaia knows everything, and so we can assume that she knew the Cyclopes would hand over the secret of lightning when they were set free. In this sense, the lightning bolts become a symbol of Gaia's favor for the Olympians; a big, fat, electric stamp of approval from Earth herself.
Also, after Zeus brings the war to a close with his furious energy display, he becomes the ruler of the gods, crackling thunderbolts in hand. If you continue your study of ancient mythology, you'll find that Zeus does in fact throw a few more lightning bolts throughout his career. In the meantime, just know they serve as a symbol of his authority over the other gods. Ancient kings often carried a scepter as a representation of their power; Zeus carried thunder.
Finally – you thought we were done didn't you? – ancient Greek culture viewed Zeus not just as the king of the gods, but also as the god of storms. Most ancient cultures have a god dedicated to lightning and thunder: Marduk for the Babylonians, Thor for the Norse, Indra in Hindu mythology, Xolotl for the Aztecs (the list goes one), and Zeus's lightning bolts are part of his traditional garb as the god of Thunder. The band Kiss even named a song after him; it's literally called God of Thunder.
With so much meaning wrapped up in Zeus' weapon of choice, it's no wonder that the first book in Rick Riordan's Percy Jackson series is titled The Lightning Thief.