Green’s Dictionary of Slang

1. (US black) a bed; a pillow.

1942		D. Burley N.Y. Amsterdam Star-News 7 Feb. 16: [S]tretching my fine brown frame on a softy.
1944		D. Burley Orig. Hbk of Harlem Jive 25: I bounced from my softy, gave the lilies a flip.

2. (Aus.) a roll of banknotes.

2006

R. Hughes Things I Didn’t Know (2007) 206: The Major [...] paid cash on the nail for it, without his usual haggling, from a thick ‘softy’ – a roll of notes.