There is some literature in the commutative algebra world that might be relevant, although I don't know it super-well. 

I don't think anything follows immediately from Grothendieck duality, especially for blowing up height-1 primes.

Say $R$ is a 2-dimensional and Cohen-Macaulay ring (equivalently S2).  Suppose that $p$ is an arbitrary height-1-prime. Since $R$ is 2-dimensional, the analytic spread of $p$ is at most $2$.  (See the book of [Swanson-Huneke on integral closure][1], there are some subtleties with analytic spread in the case of a finite residue field -- be careful).  We have two cases.  

<h2>The analytic spread of $p$ is 2</h2>  Probably this is the more interesting case.  Suppose there happens to be *only one* ideal $J$, with two generators such that $J p = p^2$ (this isn't as strong as it might sound, it's a bit stronger than requiring that $J$ and $p$ have the same normalized blowup).  Additionally suppose that $R_p$ is regular (maybe this is too strong).  Then [Theorem 3.1 in this paper by Santiago Zarzuela][2] proves that the Rees Algebra of the ideal is Cohen-Macaulay, and hence so is the blowup.  There is also a lot of potentially relevant stuff in [this paper of Huckaba and Huneke][3].  (You can look at the papers which cite it on mathscinet to find even more).

<h2>The analytic spread of $p$ is 1</h2>  In particular, then the blowup of $p$ is some finite integral extension of $R$ (in particular, the blowup is an affine scheme dominated by the normalization of $R$).  Since $R$ was S2, this implies that $p$ is a prime defining an irreducible component of the non-normal locus.  You want to keep the blowup S2...  I don't know in general if this is possible but I recall that some conditions for such blowups being normalizations appeared towards the end of [this paper by Greco and Traverso][4].  Does your surface happen to be seminormal?


  [1]: http://www.reed.edu/~iswanson/book/%E2%80%8E
  [2]: http://www.ams.org/journals/proc/1995-123-12/S0002-9939-1995-1286012-5/
  [3]: http://www.ams.org/journals/tran/1993-339-01/S0002-9947-1993-1123455-7/home.html
  [4]: http://www.numdam.org/item?id=CM_1980__40_3_325_0