Claiming

The act of being claimed occurs when a god acknowledges his/her demigod offspring by sending a hologram of their symbol of power to appear over their child's head.

Children of minor gods do not have their own cabins, so they are sent to live in the Hermes cabin once claimed and are probably only given small recognition due to their parent's minor status. However, as part of Percy's request to the gods, cabins are made for all of the children of the gods, including children of Hades.

Some campers are never claimed, so they also stay in the Hermes cabin along with the minor gods and the children and Hermes. This is also remedied by Percy and now all gods, weather Big Three, Olympian or minor have all claimed their demigod offspring.

It used to be a rare thing to be claimed, and took a long time to happen, unless the camper knew their god parent already or demonstrated a skill. Now it occurs on a camper's 13th birthday, anytime before it, or when they make it to Camp Half-Blood.

Poseidon's Claiming
When Poseidon claims one of his demigod or cyclops children, a trident hologram appears over their head. This happened to Percy after he healed his Hellhound wounds. It later happened to Tyson.

Aphrodite's Claiming
When Aphrodite claims her children, they are zapped. After that, they instantly look great. They also glow with a red aura. This happened to Piper.

Hephaestus' Claiming
When Hephaestus claims his demigod children, a red fiery hammer appears over their head. This happened to Leo.