Suppose $K$ is a number field. Let $p$ be the smallest prime which is the norm of some principal ideal. It follows that every class in the ideal class group of $K$ contains some ideal of norm prime $q<p$.
Example of conjecture above:
The class number of the field $K=\sqrt-271$ is $11$ and the smallest prime $p$ which is a norm of some principal ideal in $K$ is $307$. This means that every other class in the ideal class group of $K$ is generated by an ideal of norm $q$ NOT exceeding $307$. (Compare this to the Minkowski's bound).
Every number field with class number $> 1$ in which $2$ splits completely and the prime ideals above $2$ are principal is a counterexample. For a start, look at quadratic number fields generated by $\sqrt{t^2-2}$; here $t + \sqrt{t^2 - 2}$ has norm $2$. The first examples are ${\mathbb Q}(\sqrt{34})$ and ${\mathbb Q}(\sqrt{79})$.