From The Fabulous Fibonacci Numbers 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).