The shady hat is a unique, fedora-style pre-War hat in Fallout 3.
The shady hat can be obtained as a reward for doing the mirelurk observation quest as part of Moira Brown's "The Wasteland Survival Guide" missions. Place Moira's observer unit inside a nest deep in the Anchorage Memorial. To get it the Lone Wanderer must place an observer unit in a mirelurk egg nest without killing any of the mirelurks in the area.
The shady hat can only be repaired with merchants.
ps3 While wearing the Chinese stealth armor and the shady hat, changing both head and bodywear will yet retain the effects of the shady hat. [verified]