"liquid in which flesh is boiled," Old English broþ, from Proto-Germanic *bruthan (source also of Old High German *brod, Old Norse broð), from verb root *bhreue- "to heat, boil, bubble;" also "liquid in which something has been boiled" (from PIE root *bhreu- "to boil, bubble, effervesce, burn"). Picked up from Germanic by the Romanic and Celtic languages (Italian brodo, Spanish brodio, Old French breu, Irish broth, Gaelic brot).
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, but rather, with Scottish braith, from Middle English bratthe "violence, impetuosity; anger, rage" (c. 1200), which is from Old Norse braðr "sudden, hasty," from brað "haste."