"craving, longing, strong desire," 1796 [Grose], perhaps a back-formation from lecher, or deformed from a figurative use of latch (v.) in a secondary sense of "grasp, grasp on to." Or perhaps from letch (v.), a variant of leach.
"lustful man, man given to excessive sexual indulgence," late 12c., from Old French lecheor (Modern French lécheur) "one living a life of debauchery," especially "one given to sexual indulgence," literally "licker," agent noun from lechier "to lick;" also "to live in debauchery or gluttony," from Frankish *likkon or some other Germanic source, from Proto-Germanic *likkojan "to lick" (from PIE root *leigh- "to lick"). The Old French feminine form was lechiere. Middle English, meanwhile, had lickestre "female who licks;" figuratively "a pleasure seeker," literally "lickster," with -ster. In 18c. sometimes leacher (Bailey), along with leacherous, leachery.
Old English læccan "to grasp or seize, catch hold of," also "comprehend," from Proto-Germanic *lakkijanan. Not found in other Germanic languages; according to Watkins probably from PIE *(s)lagw- "to seize" (see lemma). In its original sense the verb was paralleled and then replaced by French import catch (v.). Meaning "to fasten with a latch" is mid-15c. Related: Latched; latching.