This is very interesting in the game. One would think that Nate is a descendant of Drake, especially how he is portrayed in the game. However, in Chapter 16 of the fourth game, they meet an old lady while treasure hunting in her mansion. She explains to the Drake brothers how she admired their mother's work.
As you said, their mother believed that Francis Drake had heirs, and she adventured to find out, all of which the old lady documented and researched. Upon finding this information out from the old lady, the young Drake brothers adopt this last name in order to honor/remember their mother's work.
It is an ambiguous topic, because as far as I can remember, Naughty Dog don't specify if Nathan is an actual descendant of Francis Drake. I guess it is just another intriguing element to the game, making it more interesting.
Hope this helped.