Green’s Dictionary of Slang

(US) running away from something threatening or dangerous.

1903		Eve. World (NY) 30 Sept. 12/5: When it comes down to artistic ‘con’ work a woman has got a man running for Sweeney.
1908		N.-Y. American 30 Aug. in Fleming Unforgettable Season (1981) 198: The bumps of their saturnine features indicated that they had been running for Sweeney.
1909		Eve. Star (Wash., DC) 17 Oct. 5/1: There are really only four sports in America [...] baseball, football, prize fighting and horse racing [...] those four get the circulation, and the others can ‘run for Sweeney’, [...] Nevertheless, if a horse runs last in a race, heis running ‘for Sweeney.’ If an actor plays to an ermpty house, he is playing ‘for Swqeeney’. If a reporter [is] writing a story after the paper has gone to press, he is writing ‘for Sweeney’.
1916		West Virginian 1 Mar. 8/7: Harry Sinclair will run for Congress in Oklahoma. That beats running for Sweeney in the Federal league.