The phrase “to scare the daylights out of somebody” means: To frighten extremely; to alarm intensely; to make one’s hair stand on end; to scare stiff.
The daylights here differ from those in the common saying, “to let daylight into a person.”
In that, the meaning is “light”, to make a hole in a person big enough for light to enter. But here, daylight means wits; hence, to scare a person witless.
In my own span of life someone, at first, one or an other of my brothers, has tried to scare the daylights out of me from the time I was two, and I’m sure the expression is just as well known and has been as long used in many another American family.
Some people amplify it to “the living daylights,” but it adds up to the same thing.
However, the recorders of the language have heretofore missed this expression.
