Finger comes unaltered from Old English (i.e. Anglo-Saxon). Indeed, the word is found, spelling unchanged, in most of the Germanic languages (e.g. German, Swedish, and Danish); The Dutch get by with vinger.

Each of the fingers have had their own Latin names. Some examples are given in the Aberdeen Beastiary, published in Scotland in the 14th century. (A bestiary is a collection of descriptions of all sorts of animals - some real, some imaginary - and other features of the natural or unnatural world.) From the Bestiary:

• "The [first] finger, index, is also known as salutaris or demonstratorius, the greeting or indicating finger, because we generally use it in greeting, showing or pointing."
• "The [second, middle] finger is called impudicus, lewd; it is frequently used to express the pursuit of something shameful."
• "The [third] is the ring finger, anularis, because it is the one on which a ring is worn. It is also called medicinalis, the medical finger, because it used by physicians to smear on ground-up salves."
• "The [fourth] finger is called auricularis, because we scrape our ear, auris, with it."
  

A digital display: Galileo Galilee's impudicus finger is on exhibit at the Museo di Storia della Scienza in Florence, Italy. www2.jpl.nasa.gov/galileo/finger.html