Fibonacci
He called himself Fibonacci [pronounced fib-on-arch-ee or fee-bur-nutch-ee] short for filius Bonacci.
There are a couple of explanations for the meaning of Fibonacci:
Fibonacci is a shortening of the Latin "filius Bonacci", which means "the son of Bonaccio", since his father's name was Guglielmo Bonaccio. Fi'-Bonacci is like the English names of Robin-son and John-son. But (in Italian) Bonacci is also the plural of Bonaccio; therefore, two early writers on Fibonacci (Boncompagni and Milanesi) regard Bonacci as his family name (as in "the Smiths" for the family of John Smith).
Fibonacci himself wrote both "Bonacci" and "Bonaccii" as well as "Bonacij"; the uncertainty in the spelling is partly to be ascribed to this mixture of spoken Italian and written Latin, common at that time.
Others think Bonacci may be a kind of nick-name meaning "lucky son" (literally, "son of good fortune").
Other names
He is perhaps more correctly called Leonardo of Pisa or, using a latinisation of his name, Leonardo Pisano. Occasionally he also wrote Leonardo Bigollo since, in Tuscany, bigollo means a traveller.
We shall just call him Fibonacci as do most modern authors, but if you are looking him up in older books, be prepared to see any of the above variations of his name.
