mid-14c., "something laid, placed or spread (on something else)," from cover (v.). Sense of "something which veils or screens from sight" is from c. 1400. From c. 1500 as "shelter" of any kind, later especially in hunting, "shrubbery, brush, or thickets which conceal game" (1719).
Meaning "binding or wrapper of a book" is from 1590s; that of "envelope or wrapper for a letter" is by 1748. Meaning "recording of a song already recorded by another" is by 1970, short for cover version (1966). Cover-band "band that plays only cover songs" is by 1981. Cover girl is U.S. slang from 1915, shortening of magazine-cover girl (1899).
Cover-charge is attested by 1913. The immediate sense of cover in it appears to be the old one of "plate, knives, forks, spoons, napkin, wine glasses, etc., used at the table by one person," from French couvert, literally "a cover," in the same sense; supposedly they were so called because they were originally brought together in a case.
According to contemporary publications, cover came to include table condiments and bread and butter, and c. 1910 some restaurants began to charge extra for these. ["... a smart New York restauranteur recently made a 'cover charge' of twenty-five cents for bread and butter and ice-water. Others followed." - Edward Hungerford, "The Personality of American Cities," 1913]
In this sense, cover also probably involves the banquet service use of cover for a charge which includes ("covers") everything provided with the food -- menu card, flowers, music, etc.
In recent years hotels, particularly those featuring entertainment in their restaurants, have made a so-called cover charge which includes entertainment in addition to the table service. For instance, at some of the larger hotels in New York, where there is dancing, or cabaret, or high-priced soloists, or entertainment of costly nature provided, there is a cover charge, sometimes as high as $1 the seat. ["Hotel Monthly," December 1917]