Consider for example the affine cone $X = V(X^2+Y^2-Z^2)$. It has a singularity at $0$, where we will blow it up.
As far as I know, the blow up is given by the variety $$\hat X = \{( (X,Y,Z),[u:v:w])\in \mathbb A^3 \times \mathbb P^2|~ Xv= Yu, ~Xw = Zu,~ Yw= Zv, X^2 + Y^2=Z^2\} $$
To me it seems like that $(0,0,0)\times \mathbb P^2\subset \hat X $
(if I set $X=Y=Z=0$, then i can freely chose $[u:v:w]$).
But this does not seem right to me. For example because of dimension reasons.