"young deer," mid-14c., from Anglo-French (late 13c.), Old French (12c.) faon, feon "young animal," especially "young deer," from Vulgar Latin *fetonem (nominative *feto), from Latin fetus "a bringing forth; an offspring" (from suffixed form of PIE root *dhe(i)- "to suck"). It was used of the young of any animal as recently as King James I's private translation of the Psalms, but the sense has been mainly of deer since 15c. Color use is by 1881.
Middle English faunen, from Old English fagnian "rejoice, be glad, exult, applaud," from fægen "glad" (see fain); used in Middle English to refer to expressions of delight, especially a dog wagging its tail (early 14c.), hence "court favor, grovel, act slavishly" (early 15c.). Related: Fawned; fawning.