Old English frogga "frog," a diminutive of frosc, forsc, frox "frog," a common Germanic word but with different formations that are difficult to explain (cognates: Old Norse froskr, Middle Dutch vorsc, German Frosch "frog"), probably literally "hopper," from PIE root *preu- "to hop" (source also of Sanskrit provate "hops," Russian prygat "to hop, jump"). Watkins calls the Old English -gga an "obscure expressive suffix."
The Latin word for it (rana) is imitative of croaking. Also in Middle English as frok, vrogge, frugge, and with sometimes plural form froggen. Collateral Middle English forms frude, froud are from Old Norse frauðr "frog," and native alternative form frosk "frog" survived in English dialects into the 19c.
I always eat fricasseed frogs regretfully; they remind one so much of miniature human thighs, and make one feel cannibalistic and horrid .... [H. Ellen Brown, "A Girl's Wanderings in Hungary," 1896]
As a British derogatory term for "Frenchman," 1778 (short for frog-eater), but before that (1650s) it meant "Dutch" (from frog-land "marshy land," in reference to their country). To have a frog in the throat "be hoarse" is from 1892, from frog as a name for a lump or swelling in the mouth (1650s) or throat infections causing a croaking sound.