This is interesting and you were very kind.
lane swerve/
As for Delta, I would have told the woman that I truly respect her love for her mother. Then I would explain (what she probably already knows which is) that wearing such things is reserved for members of DST. Then I would have said that, if she is interested, she can pursue alumnae chapter membership and (if this woman is awesome) I would offer to write her a letter of recommendation. Chances are, given her education level and that she is a Delta legacy who is wearing her mother's Delta item, she already knows about alumnae chapter membership. But telling her is worth a try. She may not be interested in (possibly) becoming a member. If she declined pursuing membership in DST through an alumnae chapter, my response to her would change accordingly all while still being professional and respecting her as a Delta legacy who is celebrating her mother's (my Soror) memory.
As an aside, I have known college students who wore their deceased parents' NPHC GLO items. That was handled much differently because 1) these were college students; and 2) it was seen as an intentional mockery when you walk up to a group of collegiate NPHC GLO members wearing an item with a "it was my deceased father's/mother's item and now it's mine--I dare you say something" smirk on your face.
/lane swerve