Take Aunt Petunia's complaint about how her sister Lily got to go to Hogwarts. You could do a lot about such a pair of sisters. Serial numbers would come off most easily if you switched things so the excluded sister was the good one. (Could go either way, of course. Those choosing could prefer one or the other or not care. Personally, since I loathe the special snowflake theory of magic, I would have to resort to some other difference between the sisters besides ability to do magic.)
Finding Sirius's relationship to Harry interesting would require rather more. It's not like it would be anything like the same unless you had something like the crime he allegedly committed, but it would need some changes to avoid the serial numbers.
Or Remus Lupin's suffering as a werewolf. One could alter it a la Bisclavret, so that he is not in fact dangerous -- but he suffers from the reputation.