Entries linking to Copenhagen
"low in price, that may be bought at small cost," c. 1500, ultimately from Old English noun ceap "traffic, a purchase," from ceapian (v.) "to trade, buy and sell," probably from early Germanic borrowings of Latin caupo "petty tradesman, huckster, peddler," cauponari "to haggle" (see chapman). Compare, from the same borrowing, German kaufen "to buy," Old Norse kaupa "to bargain, barter," Gothic kaupon "to traffic, trade."
The sense evolution is from the noun meaning "a barter, a purchase" to "a purchase as rated by the buyer," hence the adjectival meaning "inexpensive," the main modern sense, via Middle English phrases such as god chep "favorable bargain" (12c., a translation of French a bon marché).
Sense of "lightly esteemed, common" is from 1590s (compare similar evolution of Latin vilis). The meaning "low in price" was represented in Old English by undeor, literally "un-dear" (but deop ceap, literally "deep cheap," meant "high price").
The word also was used in Old English for "market" (as in ceapdæg "market day"), a sense surviving in place names Cheapside, East Cheap, etc. To do something on the cheap "with very little expense" is from 1859. Cheap shot originally was U.S. football jargon for a head-on tackle; extended sense "unfair hit" in politics, etc. is by 1970.
German billig "cheap" is from Middle Low German billik, originally "fair, just," with a sense evolution via billiger preis "fair price," etc.
late Old English hæfen "haven, port," from Old Norse höfn "haven, harbor" or directly from Proto-Germanic *hafno- (source also of Danish havn, Middle Low German havene, German Hafen), perhaps from PIE root *kap- "to grasp" (source of have) on notion of place that "holds" ships. But it might rather be related to Old Norse haf, Old English hæf "sea" (see haff). Figurative sense of "refuge," now practically the only sense, is c. 1200.