The Nightstick is an Eileen-only weapon found in Silent Hill 4: The Room. It can be found on the third floor of the Water Prison, inside the circular observatory room, but only appears there once Henry brings Eileen into that world. It is the most powerful melee weapon for Eileen Galvin to use, greatly damaging monsters.
Compared to Eileen's other melee weapons, however, the nightstick lacks the reach needed for Eileen to finish off enemies that are knocked down, so Henry will have to do that himself, or else have her stick with the chain instead.