Green’s Dictionary of Slang

toss n.1

[toss (off) v.]

1. [late 19C+] in fig. use, something of infinitesimal or zero importance; usu. in phr. not give a toss.

2. [1940s+] nonsense.

3. see toss-off n.

In phrases

give a toss (v.) (also care a toss)

[1950s+] to care; usu. in negative use, e.g. who gives a toss.