Olive O-live, O.
An Italian study found that women whose diets included a lot of olive oil had a 30% lower risk of ovarian cancer. The reasons are unclear, but the healthy fats in the oil may help suppress genes predisposed to causing cancer.
egg (n.) mid-14c., from northern England dialect, from O.N. egg, which vied with M.E. eye, eai (from O.E. æg) until finally displacing it after 1500; both are from P.Gmc. *ajja(m) (cf. O.S., M.Du., Du., O.H.G., Ger. ei, Goth. ada), probably from PIE *owyo-/*oyyo- ”egg” (cf. O.C.S. aja, Russian jajco, Breton ui, Welsh wy, Gk. oon, L. ovum); possibly derived from root*awi- ”bird.
olive (n.) c.1200, “olive tree,” from O.Fr. olive ”olive, olive tree” (13c.) or directly from L. oliva ”olive, olive tree,” from Gk. elaia ”olive tree, olive,” probably from the same Aegean language (perhaps Cretan) as Armenian ewi ”oil.” Applied to the fruit or berry of the tree in English from late 14c. As a color from 17c. Olive branch as a token of peace is from early 13c.
Both words, although geographically and linguistically unrelated ultimately come from a ‘uwi’ sounding word, egg – AWI, and olive – EWI.
See the rest of the series on the Doctrine of Signatures here