broth (n.) Look up broth at Dictionary.com
Old English broþ, from Proto-Germanic *bruthan (cognates: Old High German *brod), from verb root *bhreue- "to heat, boil, bubble; liquid in which something has been boiled" (source also of Old English breowan "to brew;" see brew (v.)). Picked up from Germanic by the Romanic and Celtic languages.

The Irishism broth of a boy, which is in Byron, was "thought to originate from the Irish Broth, passion -- Brotha passionate, spirited ..." [Farmer], and if so is not immediately related.