brad n.

"stiff, coarse hair of certain animals," especially those set along the backs of hogs, Old English byrst "bristle," with metathesis of -r-, from Proto-Germanic *bursti- (source also of Middle Dutch borstel, German borste, Danish börste), from PIE *bhrsti- from root *bhars- "point, bristle" (source also of Sanskrit bhrstih "point, spike"). With -el, diminutive suffix. Extended to similar appendages on some plants and insects.

1530s, "to poke with a stick," of uncertain origin; possibly [Barnhart, Century Dictionary] a variant of brod, from Middle English brodden "to goad," from Old Norse broddr "shaft, spike" (see brad), or perhaps imitative [OED]. Compare dialectal prog "pointed instrument for poking" (1610s), also as a verb, "to poke about." 

Figurative sense of "mental incitement or instigation" is by 1871. Related: Prodded; prodding.