The fasia board width depends on the construction of the eves. In most homes that will be six inches (6").
I have seen 3 inches from time to time, but I did not consider them to be properly installed, or were on a shed of or bilding of that type.
If you are hanging gutters on the fascia, you will probably want ot go with something a bit larger than 3 inches. 4 or 6 inches would be better.