Answered already, but I think technically if you're total insomniacs and never have to sleep, you technically don't even need a wilderness permit. How the heck do they really know if someone just slept without a sleeping bag or tent? Some people start their hikes at 3 AM, while others get back with a flashlight past midnight.
There are very few NPS areas where one needs a permit to day hike. The one we know of is the final ascent up the Half Dome cables, which is to limit the number of people. Another one is the Fiery Furnace in Arches, where people need to watch an orientation video. They have an issue with hikers getting lost or hurt.
The only place I've hiked where routine day hiking requires a permit is Desolation Wilderness near Tahoe. There it's a no-quota, no-fee self-issued permit. You write your names on the permit, your intended destinations down, drop the top copy in a box, and attach the permit to the group leader's pack or person. I'm not exactly sure what's the point, but they've done it this way for years. I think I've filled it out five times, and the form changed at least once.
