The Pepsi-Cola Company was founded in 1902, only four years after the elixir's invention in New Bern, North Carolina, by drugstore owner Caleb D. Bradham! The concoction was intentionally created to compete with the many kola nut-derived
drinks on the market at the time, especially the Coca-Cola brand. Bradham enjoyed mixing special fountain drinks for his friends at the soda fountain in his drugstore. Originally called "Brad's drink" by Bradham's friends, he renamed it "Pepsi-Cola" for its alleged pharmaceutical qualities. Pepsi derives its name from the ailment it was advertised to relieve: dyspepsia. In modern times, dyspepsia is called indigestion. Early ads for Pepsi-Cola tout this benefit: "Pepsi-Cola: At Soda Fountains. Exhilarating. Invigorating. Aids Digestion."

Sources:

Milward W. Martin, Twelve Full Ounces (New York: Holt, Rinehart & Winston, 1962);

Adrian Room, Dictionary of Trade Name Origins (London: Routledge & Kegan Paul, 1982).