From [*The Fabulous Fibonacci Numbers*][1] by A.S. Posamentier and I. Lehmann (Prometheus Books, New York, pp. 17-18):

>Leonardo Pisano - or Leonardo of Pisa, Fibonacci - his name as recorded in history, is derived from the Latin "filius Bonacci," or a sun of Bonacci, but it may be more likely derive
from "de filius Bonacci," or family of Bonacci.
 
> It is unclear who first used the name Fibonacci; however, it seems to be attributed to Giovanni Gabriello Grimaldi (1757-1837) at around 1790, or to Pietro Cossali (1748-1815).


  [1]: http://books.google.co.uk/books?id=-O4ZAQAAIAAJ&q=The+Fabulous+Fibonacci+Numbers&dq=The+Fabulous+Fibonacci+Numbers&hl=en&ei=jETQTIS3LY2-4garhLmwBg&sa=X&oi=book_result&ct=result&resnum=1&ved=0CC8Q6AEwAA