It's not too old a song for Steve Moffat to have heard it at numerous weddings--and of course, the lyrics fit well enough, so it's one that might have lept to his mind.

That's a point.  I thought using "Madonna" in the drunken game seemed as if it was a different age group too - there are more current female celebrities whom Sherlock would see in the news more.   I'm sure there's something else like that.  Maybe it's because the writers are not that far off the actors in age that it gets mixed up.

It's placed so significantly, though and heard so clearly that I think it has to be deliberate, doesn't it? This is the scene that's supposed to be like the Doctor Who episode and I wondered if that was significant - I did check the release dates but December 1963 wasn't released until a couple of years after The Green Death. 

The line about not knowing her name fits.

Madonna makes sense to me, she was big in the 80s 90s and 00s. Even people with no interest in music at all have heard of her. Never heard that other song though. 

Why would it be special to John and Mary?  I know there could be any reason - it was playing on the radio when they met, or on their first date, or whatever.  But there must be a reason for the writers to have chosen it as a special song for them, as it features so prominently.  Or am I just reading too much into it?

Sherlock claims he just chose the name Madonna from the papers ... of course, she could be in today's papers, but ... I can't explain, it jarred, as if a name had been chosen from the wrong generation.  Something else made me feel the same, not in that episode I don't think, but it's not coming to mind now.  

After sleeping on it, I still don't know about December 1963, but I have some thoughts on the choice of Madonna!

Madonna usually refers to Mary, mother of Jesus.   According to some Christian belief, Mary was conceived free of original sin - the immaculate conception.   And "Mary Morstan" had an "immaculate conception" five years previously. 

Sherlock chooses that name, then says he doesn't know who she is, and waves it away as something in the papers.  But maybe it shows something about his subconscious.  John can't guess it because Sherlock can't tell him anything about Madonna.  Who is she?

So I wonder if this is foreshadowing of what's going to happen with Mary.  The immaculate conception/the false identity.  The pregnancy, obviously.  The presentation of Mary as a virtuous woman (another meaning of madonna).   Maybe the choice of Madonna was much more clever than I thought!

Which makes me think even more that December 1963 has got to be significant too.

Meh, I just think Steve Thompson thought of a random famous person's name. I don't know exactly how old he is, but I think it would fit his age. 

Hey, don't spoil it for me!  I was so excited when I woke up this morning thinking that Mary was the immaculate conception!   It's perfectly normal to examine details like this.  Yes, it is.

@Liberty and silverblaze
I think when it came to picking a name Sherlock would have chosen for the Rizla game the writers referred to Benedict's encounter with Madonna some time ago when she teased him about his name: "Ooh, yeah, you’re the one with the strange name!" and when he replied something like: "Yes, I am, Madonna."
http://www.benedictcumberbatch.co.uk/interviews/gq-benedict-cumberbatch-outtakes/