Hardhats / Engineering Helmets.

[Winterous]

When you pick up a Hardhat or Engineering Space Helmet, the little icon in the top-right to toggle the light shows up.
But when they're no longer on your person, even if they're now being worn by another person, the icon stays there!
GAH, THE OCD!

[Rob Stark]

This happens to me all the time, even with the backpack sometimes too.

[Spy_Guy]

They also serve as a free flashlight, making the dark halls of inner engineering that much cosier.

[Winterous]

They also serve as a free flashlight, making the dark halls of inner engineering that much cosier.

I don't think it always works that way, it seems to me that sometimes the actual light source sticks with your character, but sometimes it stays on the hat.
It's really weird.