I have a pjo/tsoa au idea
What if Achilles became a god like Thetis wanted him to? What if he gave Patroclus some of that immortality? What if neither of them died? So heres what would happen if they're minor gods in the riordonverse
They would only get cabins after the first Percy Jackson series, and the cabins would be right next to eachother, probably decorated similarly. The Achilles cabin would probably have 20-25 campers each century, and the Patroclus cabin would have 15-20 campers. The campers might consider the other cabin their siblings, but it wouldnt be uncommon for them to date. Some of the Achilles campers might have sea related powers as Achilles was born from Thetis, a sea nymph. Both cabin's campers would be skilled fighters.
The Achilles cabin and the Apollo cabin would have a rivalry. They'd either be wary or even agressive to the other. Main part of the rivalry is the mastery of the lyre, as the Achilles kids would naturally be skilled lyre players, and the Apollo kids are musically skilled too.
The Patroclus campers would powerless in a way, but probably be smart planners of battle strategists. I think they'd get along with the Athena campers.