There is some evidence from characters that $H^4(M,\mathbb{Z})$ contains $\mathbb{Z}/12\mathbb{Z}$.  In particular, the conjugacy class 24J (made from certain elements of order 24) has a character of level 288, and the corresponding irreducible twisted modules have a character whose expansion is in powers of $q^{1/288}$.  Fusion in a cyclic group generated by a 24J element then yields a $1/12$ discrepancy in $L_0$-eigenvalues, meaning you will pick up 12th roots of unity from the associator.  If you pull back along a pointed map $B(\mathbb{Z}/24\mathbb{Z}) \to BM$ corresponding to an element in class 24J (i.e., if you forget about twisted modules outside this cyclic group) you get a cocycle of order 12.  This is the largest order you can get by this method - everything else divides 12.  I don't know how the cocycles corresponding to different cyclic groups fit together.

I don't know if you've seen Mason's paper, [Orbifold conformal field theory and cohomology of the monster][1], but it is about related stuff.  I don't understand how he got his meta-theorem with the number 48 at the end, though.

As far as implications or meaning of the cocycle, all I can say is that the automorphism 2-group of the category of twisted modules of $V^\natural$ has the monster as its truncation, and its 2-group structure is nontrivial.  I've heard some speculation about twisting monster-equivariant elliptic cohomology, but I don't understand it.  If you believe in AdS/CFT, this might say something about pure quantum gravity in 3 dimensions, but I have no idea what that would be.

**Update Nov 2, 2011:** I was at a conference in September, where G. Mason pointed out to me that $H^4(M,\mathbb{Z})$ probably contains an element of order 8, and therefore also $\mathbb{Z}/24\mathbb{Z}$.  I believe the argument was the following: there is an order 8 element $g$ whose centralizer in the monster acts projectively on the unique irreducible $g$-twisted module of the monster vertex algebra $V^\natural$, such that one needs to pass to a cyclic degree 8 central extension to get an honest action.  Rather than just looking at $L_0$-eigenvalues, one needs to examine character tables to eliminate smaller central extensions here.  Naturally, like the claims I described before, the validity of this argument depends on some standard conjectures about the structure of twisted modules. 

It seems that the relevant group-theoretic computation may have been known to S. Norton for quite some time.  In his 2001 paper *[From moonshine to the monster][2]* that reconstructed information about the monster from a revised form of the generalized moonshine conjecture, he explicitly included a 24th root-of-unity trace ambiguity.  I had thought perhaps he just liked the number 24 more than 12, but now I am leaning toward the possibility that he had a good reason.

  [1]: http://www.newton.ac.uk/programmes/NST/Mason.pdf
  [2]: http://books.google.co.jp/books?id=GYmU_9EKQCEC&lpg=PA163&pg=PA163#v=onepage