large, long-tailed, stalk-eyed, 10-legged marine shellfish (Homarus vulgaris), early Middle English lopster, lopister, from Old English loppestre "lobster," also "locust," a corruption of Latin locusta, lucusta "locust, grasshopper," which is of unknown origin. De Vaan writes that "The only word similar in form and meaning is lacerta 'lizard; mackerel', but there is no common preform in sight. ... [T]hey could be cognate words in the language from which Latin borrowed these forms."
The change of Latin -c- to English -p- (and, from late 14c., to -b-) is unexplained; perhaps it is by influence of Old English loppe, lobbe "spider." The ending seems to have been altered by the old fem. agent noun suffix (preserved in Baxter, Webster, etc.; see -ster), which approximated the sound of Latin -sta.
OED says the Latin word originally meant "lobster or some similar crustacean, the application to the locust being suggested by the resemblance in shape." Trilobite fossils in Worcestershire limestone quarries were known colloquially as locusts, which seems to have been the generic word for "unidentified arthropod" (as apple was for "foreign fruit"). Locusta in the sense "lobster" also appears in Old Cornish legast and French langouste (12c.), now "crawfish, crayfish," but in Old French both "lobster" and "locust" (a 13c. psalter has God giving over the crops of Egypt to the langoustes).
As slang for "a British soldier" since 1640s, originally in reference to the jointed armor of the Roundhead cuirassiers, later (1660) to the red coat, the color of a boiled lobster.
Sir William Waller having received from London [in June 1643] a fresh regiment of five hundred horse, under the command of sir Arthur Haslerigge, which were so prodigiously armed that they were called by the other side the regiment of lobsters, because of their bright iron shells with which they were covered, being perfect curasseers. [Lord Clarendon, "History of the Rebellion," 1647]