There have been 22 cycles in Red Sox history; Betts is the 21st player to do it (Bobby Doerr did it twice).
Here's the list of players who have hit for the cycle for the Red Sox:
Mookie Betts, Aug. 9, 2018
Brock Holt, June 16, 2015
John Valentin, June 6, 1996
Scott Cooper, April 12, 1994
Mike Greenwell, Sept. 14, 1988
Rich Gedman, Sept. 18, 1985
Dwight Evans, June 28, 1984
Fred Lynn, May 13, 1980
Bob Watson, Sept. 15, 1979
Carl Yastrzemski, May 14, 1965
Lou Clinton, July 13, 1962
Bobby Doerr, May 13, 1947
Ted Williams, July 21, 1946
Bob Johnson, July 6, 1944
Bobby Doerr, May 17, 1944
Leon Culberson, July 3, 1943
Joe Cronin, Aug. 2, 1940
Moose Solters, Aug. 19, 1934
Roy Carlyle, July 21, 1925
Tris Speaker, June 9, 1912
Patsy Dougherty, July 29, 1903
Buck Freeman, June 21, 1903