HEADCANON #1:
One time Jason and Piper disappeared on the Argo II. The other seven were forced by Coach Hedge to look for them. Of course, it was Frank who found them in the engine room making out, and he silently cursed to himself and said, "Shoot! Not again!" Percy and Annabeth were very amused to hear this until Coach Hedge sentenced all the couples on the ship to cowbells, angering innocent Hazel and Frank who claimed they would never do anything like that of the other couples. (Behind them, Leo was wiggling his eye-brows.)
HEADCANON #2:
It was really hard for Percy to sleep when he got back to Camp Half-Blood after facing Gaea. He remembered having Annabeth by his side everynight on the Argo II. She would be there for him when he woke up in sweat from the nightmares of Tartarus...and he was there for her. Now she had her cabinmates and he had no one. Once Annabeth found out he wasn't sleeping, she would sneak over to his cabin and spend the night just holding him...so he could finally get some rest, free of fear.
HEADCANON #3:
On New Year's Eve, Leo was feeling up to some of his old tricks. Since all the stuff with Gaea had been resolved, Camp Half-Blood was celebrating the New Year with Camp Jupiter! Noting Rachel and Octavian's utter distaste for each other, Leo couldn't help but string up some enchanted mistletoe (left over from their Christmas celebration) above where they would meet to announce the upcoming fortune for the New Year. Everything went according to plane at first, until Rachel and Octavian began to kiss passionately in the middle of the announcement of the next prophecy, shocking everybody. (Rachel turned bright pink, but that didn't even compare to the shade of red Octavian turned!)
HEADCANON #4:
Annabeth (being a long-time fan of Doctor Who even before she left her father) could barely bring herself to watch the last episode with Eleven. With some coaxing from Percy, and a whole lot of participation on the seven's part, they had her sitting down on the Argo II watching the projection and bawling her eyes out into Percy's shoulders once the episode was over. Everyone wondered how anyone could get that emotionally attached to something seemingly meaningless when Annabeth explained something. She had been able to think about the Doctor and his marvelous adventures and pretend she was living in them whe she was alone at only seven years old. It became a game to tell herself in dire situations, "It's okay. Oh, the Doctor will save me. I'll be alright. This time...nobody dies..."
HEADCANON #5:
It was Piper who walked in on Leo having a mental breakdown. He had disappeared into the engine room for a few hours now, and everyone was beginning to grow worried. He was in the corner, grease smudges on his face, bawling and mumbling into his knees, "What if I don't find her? What if I'm too late?" Piper went over to him and wrapped her arms around him like any good friend would. When he had calmed down enough to go to sleep, Piper went to the upper decks to talk to Jason. She told him, "Whoever she is, he must love her an awful lot."