Old English hara "hare," from Proto-West Germanic *hasan- (source also of Old Frisian hasa, Middle Dutch haese, Dutch haas, Old High German haso, German Hase), of uncertain origin; possibly the original sense was "gray" (compare Old English hasu, Old High German hasan "gray"), from PIE *khas- "gray" (source also of Latin canus "white, gray, gray-haired"). Perhaps cognate with Sanskrit sasah, Afghan soe, Welsh ceinach "hare." Rabbits burrow in the ground; hares do not.
þou hast a crokyd tunge heldyng wyth hownd and wyth hare. ["Jacob's Well," c. 1440]
"dried berries of the pepper plant," Middle English peper, from Old English pipor, from an early West Germanic borrowing of Latin piper "pepper," from Greek piperi, probably (via Persian) from Middle Indic pippari, from Sanskrit pippali "long pepper." The Latin word is the source of German Pfeffer, Italian pepe, French poivre, Old Church Slavonic pipru, Lithuanian pipiras, Old Irish piobhar, Welsh pybyr, etc.
Application to fruits of the Capsicum family (unrelated, originally native of tropical America) is from 16c. To have pepper in the nose in Middle English was "to be supercilious or unapproachable."