I think Mysterious Melody would work a lot better if it was more similar to Secret Santa events in real life, in that senders would know who they were sending to, but it would not be their MMs. Their own MMs would be revealed to them at the end of the 24 hours. Not only would this make gift-giving more personal and easy, it would also help keep users accountable for whether they choose to send gifts or not. I don't think that there is much to lose with this arrangement, other than the possibility of bad MMs being questioned by disappointed recipients, but I think this system would highly discourage people from neglecting their recipients. I personally would also like to be able to give my MMs a proper thank-you at the end of the day.

This could lead to so much harassment, please don't.
This would turn out so bad it isn't even funny.
Watching people foam at the mouth when they claim that the other person hasn't spent enough on return presents is bad enough. To see that rage directed at a specific person would be pretty awful.