Rust-colored Egg

The Rust-colored Egg is one of two eggs found near the end of Silent Hill 2. James Sunderland collects it after defeating the Pyramid Head duo in the first floor lobby of the Otherworld Lakeview Hotel, whence he came to terms with his guilt.

The connection the eggs have to Pyramid Head and James isn't specified in-game, but they are both used to unlock the doors leading to the final boss. Once both are inserted into the lobby doors, they will unlock and grant access to the final hallway.

Symbolism
The rust-colored egg could symbolize Mary Shepherd-Sunderland, as she could be represented as "old", "diseased", "ugly", and "dead", while the scarlet egg could represent Maria, who is "new", "healthy", "beautiful", and "alive". In the same vein, it's possible that the rust-colored egg represents the old, congealed blood of Mary's death by James's doing, as opposed to the recent, fresh blood of Eddie Dombrowski's.