Why not do it another day?
This is a good example of how JWs are taught to discuss things with outsiders and attempt to make their odd beliefs seem reasonable. Your mother was being deceptive even if she didn't know that was what she was doing. If you said "Ok, why not another day? Let's pick a day and we'll do it" she would be forced to back track because the day isn't the issue. The Watchtower just doesn't want their followers celebrating the birth of Jesus. I think the Watchtower views Jesus as someone that has to be kept in his place. Just look at what they have done to the simple celebration of Christian communion. They've turned it into a yearly event where JWs gather together and essentially thumb their noses at Jesus refusing the offer.
Another reason is that they don't want their follower participating anything that has a community spirit and would bring them closer to none JWs. Cults are all about separation and control.