I may be misunderstanding something but this question does not seem to have anything to do with Hironaka's desingularization. You are asking if you can resolve the indeterminacy of a rational map, right? If this is the question, then you can do it with finitely many blow-ups at points. 

Suppose you have a rational map $f$ from a smooth projective surface $S$ to $\mathbb{P}^{n}$ given by some linear system $V \subset |D|$. Without loss of generality you may assume that the image of $f$ is non-degenerate, i.e. is not contained in any hyperplane. In that case the base locus of $V$ is at most finite. Now take a base point of $V$ and blow it up $g : \widehat{S} \to S$. The pullback $g^{\*}V$ will consist of divisors in $|g^{\*}D|$ which contain a positive multiple $nE$ of the exceptional divisor $E$. So $g^{\*}V$ has a fixed component and therefore is of the form $\widehat{V} + nE$, where $\widehat{V} \subset |g^{\*}D - nE|$. The linear system $\widehat{V}$ gives a rational map $\hat{f}$ from $\widehat{S}$ to $\mathbb{P}^{n}$ which has no fixed component and coincides with the original $f$ on $\widehat{S} - E$. Now set $\widehat{D} = g^{\*}D - nE$. Note that the fact that $\widehat{D}$ has no fixed component implies that $\widehat{D}^{2} = D^{2} - n^{2} < D^{2}$. If $\hat{f}$ has no base point, then we are done. If it does have base points, then we can repeat the process. Since the degree of the linear system decreases on each step the process will terminate.