What does the Betrayer give you?
Solution 1:
This seems to be revealed later in the game. It's a special dagger.
"After providing a battery to the Celestial Locator, the Betrayer also gives the Slayer a special dagger to destroy his son's heart that controlled the Icon of Sin in the hopes of giving peace to his son's soul."
It's a story item used once again in a cutscene, there's no way for the player to interact with it.
source (spoilers)