According to "Online Etymology" paddle first entered the English language in 1407 and originally had the meaning of "a short spade". Then a kind of oar to steer a boat with. It’s more popular meaning as a flat board to beat clothes ( or schoolboys) with dates from 1828 and is apparently American in origin.
The word bears some resemblence to French poêle meaning a "frying pan" and the two words do have a common origin in a Medieval Latin word ‘padela’ which Online etymology says is probably a variation of Classical Latin patella (pan; plate). A cognate is also found in Occitan (Provencal) padèla also meaning "frying pan" and its Canadian French variant poêlon. ::)
