The most likely root would be the Arabic word al-fil, the elephant.
[Sansk. Hasti,] Pers. pil, Arab. al-fil --> Sp. alfil, It. alfiere (meaning ensign), [Fr. fou (=joker)] ?--> Eng. elephant
Many words that trace their roots to Arabic incorrectly carry the definite article al along into the destination language: al-zebr --> algebra.
When chess came to Persia from India, the Sanskrit name (Hasti –note by me) was translated to pil, and when chess came to the Muslims from Persia, the move had not changed, and the only changes to the name were made to suit Arabic phonology. The name thus became fil and then alfil (prefixing the Arabic definite article, al). The names sometimes changed even more when chess eventually reached Europe, but eventually started to refer to the modern bishop rather than the alfil. … alfil is still the name of the Bishop in Spanish and some other European languages …
Bodlaender on piececlopedia:
Golombek also points out that the English word elephant was also borrowed from another language. Although I can't document the connection, elephant does sound similar to alfil. Perhaps it has its roots in the name for a Chess piece. [Golombek, Harry. Chess: A History, 1976.]
Meta: Will try to add further if I find more details along the way.