I know MM needs a little love. There are some nice suggestions and I didn't read them all so if I repeat some ideas... sorry.
I do like the restart button after 12 hours, if your MM partner is a no show from the get go.
I think if you rate a MM partner, there could be a series of reason why you gave that partner their score, similar to when you send out an item to your MM partner. 1 = not in the spirit, 2 = lovely and sweet, 3 = generous, blew my mind.
I know that I start off slow, with a 50 point token to see where we go from there and then I fairly match their stride. But after the first 3 nights, I peter out and stop playing. LOL. I think if you are TOO generous, they feel bad and then stop. That is my experience.
But, yes, this part of Lumi needs a little TLC.
[edit] Thank you for entering the Mysterious Melody activity! You have a partner, and you can now send items to them from your inventory on the front page!
Thought I'd give it a try again. Let's see what happens.

I also think there needs something to be done. It is just sad if you get in first and then wait, then you get the message someone joined and you think YAY!, Start up gifting and there comes... nothing! Not a single thing! I don't care if my MM sends me few or cheap items. but just plain nothing is just not in the spirit of the time, especially if you knew they have joined after you and they should therefor be there.
I'd even be for a bit stricter sanctions. If someone enters for lets say three days in a row without giving anything, the person should get a ban for a while since they aren't in the spirit anyway.