I guess the tradition of it being the man's name passed on means that's why there is no common surnames with *dottir as it is with *son? (Not sure what the english version for a daughter is).
Had some Icelandic friends in school (which still has patronymic names, moved here after they were born), and it was for them somewhat problematic at times that the siblings had different surnames (Björnsdottir and Björnsson), as people don't assume they're family, and especially not that the parents both had different surnames again. Like school pickup with a teacher not knowing the situation.
And the newer meaning is probably why almost no Richards go by "Dick" anymore.