The summer solstice is the summertime meeting between the Greek gods of Olympus. The summer solstice is the longest day of the year and is when the Greek gods' magic/powers are at their strongest. It occurs every year on June 21st and is when the Tropic of Cancer is closest to the sun.
After Percy Jackson, the son of Poseidon, survives the hellhound's attack, Poseidon claims him. This causes discord among the gods of Olympus, as this is proof that Poseidon had not honored the pact he had made with his two brothers.
After Zeus' lightning bolt is discovered to be missing, Zeus suspects Poseidon used Percy to steal it and demands he returns it by the summer solstice. Poseidon, in turn, wants an apology for being suspected by that same date.
A quest is given to Poseidon's son Percy, to find the missing lightning bolt and return it in time in order to stop a war amongst the gods. If he failed, it would be the Trojan War all over again, except on a much larger scale. Nature would be at war with itself, and it would be many years of battle until the winner came to victory, nothing could be sure about what the world would be like then.
Percy succeeds in his quest as he recovers Zeus' lightning bolt by the summer solstice and reveals to Zeus and Poseidon that the thief was Luke Castellan, son of Hermes. Poseidon meets his son formally at the summer solstice.
After Leo Valdez announces that he must build the Argo II, Chiron mentions that they must sail by the summer solstice, since that's when the gods powers are strongest and summer winds will be the easiest to navigate. No longer than that date or it would be too late.
While on the quest to free Thanatos, Percy has a dream of Piper McLean, Jason Grace, and Leo Valdez, as Leo fixes the Argo II. Piper mentions that they were supposed to leave on that day, since it was the solstice. Since the events of their quest happened approximately around the June 20th-24th, it's most likely Piper was talking about the summer solstice.