Elliptic curves for ECDSA
It is easier to generate a point with order $n$ than to find out the order of a random point:


Generate a random point $G&#39;$ (generate random $x$ and solve for $y$)
Compute $G = hG&#39;$ (multiply by cofactor)


This is guaranteed to generate a point $G$ with order either $n$ or $1$ (the point at infinity). The chance of generating the point at infinity is negligible, but you can check for it and regenerate $G$ if you want.

This procedure is described in section 3.1.3.2 "Point Selection" in SEC1v2, where it computes a generator from a seed $S$. It is more convoluted since it generates points verifiably at random.