Given the nature of the mother character, don't you think it ought to qualify?
The "with name" part of the requirements seems to me just to make sure that these are significant characters, not throwaway, nameless cardboard cutouts. The mom is a great character, with or without a name.
Would you consider Eastwood in The Good, the Bad, and the Ugly unqualified for a "real character" just because he is never given an authentic name?
One of may fav movies of all time.