Well, yes. I want to hand-wave that away, since it seems implicit to me that that’s what is being assumed with this type of puzzle (I’ve seen countless versions of these logic games), but you’re correct. That assumption should be explicitly stated.
Whoops…that should read that we know Ben can’t have any M that corresponds to the unique Ns.
And how did Bill enter the conversation? 
I’m not sure this is giving us enough info. Bill (assuming you mean “Ben”) will never have enough information to come up with any conclusion on his own. So Ben saying he doesn’t know doesn’t give us anything we didn’t know to begin with. You may as well skip starting the conversation with Ben, unless he says something interesting like “I don’t know, but I know Mark doesn’t know, either.”
When Mark says “I don’t know either,” that tells us he doesn’t have a unique N, so we eliminate 7/6/70 and 2/12/70. We now have:
4/30/70, 5/3/70, 8/3/70
4/6/70
1/9/70, 5/9/70
1/12/70, 8/12/70
Bill saying “yeah it’s a mystery” means that he’s not holding an M that leaves us with one N, so 4/6/70 is eliminated. If Bill here had said “Oh, I know it,” it would be 4/6/70. But in either case, the first statement is unnecessary. If the conversation started with Mark saying “I don’t know the date” and then Bill saying “Oh, I know it now” then Mark would respond “So do I” as the only possible route to that conclusion is with the date 4/6/70, with Bill holding the 6.
So I don’t see how we can narrow farther, given what you’ve given us.
I haven’t ridden the train of logic out to its conclusion, but I’m wondering if by this statement you’re really trying to say something like “I don’t know, but I can’t ensure that Mark doesn’t know,” (or whatever the logically sound way of phrasing it is) to point us to an M = 6 or 12.
Whoops, I meant Ben. He was using an alias to avoid the FBI.
I think it works, but check your dates; you have 4/30/1970 instead of 4/3/1970
The dates I have shouldn’t make a difference that I see. That’s just a typo.
Bill is Mr Tan’s first name. 
Suppose Mark holds 4/xx/xxxx. Then he knows that Bill holds xx/3/xxxx or xx/6/xxxx. If Bill holds xx/6/xxxx then he will know that Mark holds 4/xx/xxxx as soon Mark admits that he doesn’t know the date. When Bill admits that he doesn’t know, then Mark knows that Bill must hold xx/3/xxxx, so the date must be 4/3/xxxx
OK, now do the same, and suppose Mark holds 1/xx/xxxx. What I’m saying is, there’s not a unique solution that we (not Mark or Bill) can figure out given the information.
[spoiler]Suppose the date is 1/9/1970. Then when Mark says he doesn’t know, Bill/Ben knows the date immediately, because Mark must hold either 1/xx/xxxx or 5/xx/xxxx.
If the date is 1/12/1970, then Bill knows Mark’s hand must be 1/xx/xxxx, 2/xx/xxxx, or 8/xx/xxxx. When Mark says he doesn’t know, then Bill can eliminate 2/xx/xxxx, leaving 1/xx/xxxx and 8/xx/xxxx. However, when Bill now says he doesn’t know either, Mark still doesn’t know if the date is 1/9/1970 or 1/12/1970.
Of course, he would figure it out on the next go-round, so we could make the question more difficult that way, but the only way Mark knows at this particular point is if he holds 4/xx/xxxx and Bill holds xx/3/xxxx[/spoiler]
N/M
Yeah, that does seem to work now. So you can basically reduce it to:
Mark: I don’t know
Ben : I don’t know
Mark: Now I know.
The first part of my spoiler is wrong. Suppose the date is 1/9/1970 or 1/12/1970. Then the series of comments don’t lead to a resolution, so it can’t be a solution
Yeah, there’s only one date with that line of conversation that makes sense (as far as I can see), and that’s the one you had.
BTW, I love these sorts of puzzles. 
We’d have to change the dates a little, but we could have a riddle where the clues are:
Mark: I don’t know
Ben : I don’t know
Mark: I don’t know
Ben : I don’t know
Mark: Now I know!
It makes sense if the dates are in British style.
Never mind, it makes sense either way.
[spoiler]First statement tells us M={3 or 9}.
Second statement tells us N≠5.
If N=5, Mark wouldn’t know if M={3 or 9}.
Therefore, N={1 or 4 or 8}
Third statement tells us M=9.
If M=9, then N=1, and Mark would know M=9.
If M=3 and N=8, then Mark would know M=3.
If M=3 and N=4, then Mark would know M=3.
That Ben can tell N from statement 2 means M=9 and N=1.
OK, I thought at first it was 4/3/1970, but looks like it is 1/9/1970.[/spoiler]
Okay, doing this without looking at other posts. As an side, this is annoyingly designed - they should’ve made Mark/Ben’s names start with the same letter as N/M, and used D instead of N. Anyways…
Ben knows M and Mark knows N. The list is already organized by M, so let’s organize it by N:
1/9 and 1/12
2/12
4/3 and 4/6
5/3 and 5/9
7/6
8/3 and 8/12
Since there’s only one date for N=2 and N=7, it can’t be either of those two dates (since it would be apparent Mark would know if either of those were N). This means we can eliminate the M=12 and M=6 possibilities, because if Ben’s M value were one of those, Mark might know.
Thus, that leaves, five possibilities: 1/9, 4/3, 5/3, 5/9, and 8/3. Mark knows N (the first number), so it’s got to be 1/9, 4/3 or 8/3, because he still wouldn’t know if it was N=5.
By the same logic, Ben must have M=9 because if he had a 3, he still wouldn’t know.
1/9/1970
And then the guru says “I see at least one person on the island with N eyes”
…no wait, wrong riddle.