heed v.

Old English hæt "hat, head covering" (variously glossing Latin pileus, galerus, mitra, tiara), from Proto-Germanic *hattuz "hood, cowl" (source also of Frisian hat, Old Norse hattr, höttr "a hood or cowl"), of uncertain etymology; it has been compared with Lithuanian kuodas "tuft or crest of a bird" and Latin cassis "helmet" (but this is said to be from Etruscan).

To throw (one's) hat in the ring was originally (1847) to take up a challenge in prize-fighting. To eat one's hat  (1770), expressing what one will do if something he considers a sure thing turns out not to be, is said to have been originally eat Old Rowley's [Charles II's] hat.

"cautious, wary," 1540s, from heed (n.) + -ful.