Although there is a good reason that $(x,y)^2$ has a smooth blow-up.  It is a power of an ideal which itself has a smooth blowup.  See for example Hartshorne, Algebraic Geometry, Chapter II, Section 7, Exercise 7.11.  

I suspect that, on smooth surfaces, one can probably say more, via "Zariski-factorization" type ideas, but I'm not sure what the right answer would be.

Edit:  I've looked around for a good reference on "Zariski-Factorization", but I'm not sure what a good one is.  Does someone know?