You will probably have to submit signed papers at your county courthouse in person at some point to change your son’s name and pay a fee. Then you will need to go to Social Security in person to fill out and sign some more papers. In both cases you can download the forms from the Internet, but you will need to go somewhere at some point.
You might better consider that while you and your father are estranged now, you may not always be so. Family members do make up all the time. Plus, perhaps your father was named for his own father or someone else. Thus, you could say that your son is named for his grandfather or a president or famous person rather than your father. The name doesn’t change but the meaning behind it does.
In addition, unless this boyfriend is your son’s biological father, you might wish to save the new name for a child that he has fathered.