Fibonacci is famous for the number sequence that bears his name today (I am not certain, but I believe the first program I wrote in Fortran on a Univac 1100/60 was for Fibonacci squares) but the Plus magazine team says he would be surprised by that; rather being famous for the famous sequence 0, 1, 1, 2, 3, 5, 8, 13, ... he might expect to be remembered for helping to popularize a modern number system in a Latin-speaking world.
And how to figure out breeding rabbits. One of the mathematical problems Fibonacci discussed in his work Liber Abaci was how fast rabbits could breed in ideal circumstances. Since rabbits can mate after a month, if the rabbits never die and that females always produces one new pair (one male, one female) every month from the second month on, how many pairs will there be in one year?
A page of Fibonacci's Liber Abaci from the Biblioteca Nazionale di Firenze showing the Fibonacci sequence in the box on the right. Link: Plus magazine.
Now we see it in everything from biology to the Golden Ratio. And his work made people want to learn numbers the way he knew them, and that was the end of Roman numerals (outside movie copyrights anyway).
The life and numbers of Fibonacci by R.Knott and the Plus team