DEAR MISS MANNERS: I have a modern technology dilemma. My mother FaceTimed me the other day to see my daughter eating (she's 5 months old). My boyfriend was sitting next to me on the couch, barely in view of the camera.
Neither my mother nor he acknowledged the other. Whose responsibility was it to say "hi" first? My mother was upset that he didn't say anything, but she also didn't say anything.
GENTLE READER: Etiquette has such a rich history that when a new question arises, Miss Manners fishes into the past for precedent.
Here the crucial point is whether the situation you describe is basically a visit, although a virtual one, or instead a telephone call, but a visual one. If your mother were visiting your house, anyone obviously present would be required to greet her. If she is telephoning you, others present should not chime in unless specifically invited.
Miss Manners would consider it sensible to go with the telephone analogy, as your mother intended to visit you and your daughter. But this would require the gentleman to keep out of sight, and "barely in view" sounds as if he ought to move over.