The summer solstice is the summertime meeting between the Greek gods of Olympus. The summer solstice is the longest day of the year and it is the time when the Greco-Roman 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, survived the hellhound's attack, Poseidon claimed him. This caused discord among the gods of Olympus, as this was proof that Poseidon had not honored the pact he had made with his two brothers.
After Zeus' lightning bolt was discovered to be missing, Zeus suspected that Poseidon had used Percy to steal it and demanded that he return it by the summer solstice. Poseidon, in turn, wanted an apology for being suspected by that same date.
A quest was 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 take many years of destructive battle until the winner came to victory, and there is no prediction as to what the world would be like then.
Percy succeeded in his quest as he recovered Zeus' lightning bolt by the summer solstice. He revealed to Zeus and Poseidon that the thief was Ares, and hinted that Kronos would have manipulated him. Poseidon met his son formally for the first time 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.