Green’s Dictionary of Slang

tit n.3

also tit-end
[? tit n.1 (2) or tit n.2 (1)]

[mid-19C+] a fool; thus look an absolute tit, look a right tit, to appear a total fool, thus the school ‘witticism’, I feel a right tit.

In phrases

big tit (n.)

[1960s–70s] (US) an important person, or one who thinks they are.

make a tit of oneself (v.)

[2000s] to make a fool of oneself.

simple-tit (adj.)

foolish.

stand there like a tit in a trance

[20C+] said of one who is lost in thought, abstracted.

tit about (v.) (also tissfatart about, tit around)

[1940s+] to play around, to waste time, to act in a trivial, pointless manner.