*not entirely my theory*

Katniss's father 'died' in a mine explosion, right?

Prim is good at healing. Katniss is good at archery.

They both like music and singing.

Her father was secretly Apollo.

Apollo is the god of archery, music, and healing. He's also the god of the sun, and when Katniss was about to go into the Games, I think one of her costumes was sun- related.

He used the excuse 'he died' so he could leave unsuspiciously. Maybe he caused the explosion.

Katniss's mother knew, but she couldn't tell anyone.

They are demigods.