Come up with a better name
Fog Burner makes no sense, so let's do better. Then we can make whatever icon is appropriate!
Random thought: how about “Curfew” for an app name? You're breaking your Energy Savings curfew by keeping the display on longer than normal.
Some additional ideas:
Vigil/Guard (shield, eye) -- but I think this could also imply anti-virus or security RedEye (eye) Nocturnal/NightOwl (owl, moon) Something like "Firefly" or "Glowworm" might be cute too.
I actually quite like “Glow Worm”, and “Red Eye” implies both caffeination and keeping awake!
Red Eye could be the smartest one though. Glow Worm could sound like a game?
Not that this is a democratic process or anything, but I like NightOwl a lot.