"handled instrument to strike the ball in tennis, etc.," c. 1500, probably extended from earlier racket "tennis-like game played with open hand" (late 14c.), from Old French rachette, requette, rechete, resquette (Modern French raquette) "racket for hitting; the palm of the hand," which is of uncertain origin.
Perhaps it comes via Italian racchetta or Spanish raqueta, both often said to be from Arabic rāhat, a form of rāha "palm of the hand," but this has been doubted. Compare French jeu de paume "tennis," literally "play with the palm of the hand," and compare tennis.
"loud, disorderly, confusing noise," 1560s, probably imitative. Klein and Century Dictionary compare Gaelic racaid "noise, disturbance," but OED says this "is no doubt from Eng."
Meaning "dishonest activity" (1785) is perhaps an extended sense, from the notion of "something going on" or "noise or disturbance made to distract a pick-pocket's victim." Or it might be from racquet, via the notion of "a game," or from or reinforced by rack-rent "extortionate rent." There also was a verb racket "carry on eager or energetic action" (1753), and the gangster sense might be via the notion of "exciting and unusual." Weakened sense of "way of life, one's line of business" is by 1891.