Potential function binary heap extract max O(1)
Try the following:

The weight $w_i$ of an element $i$ in the heap $H$ is its depth in the corresponding binary tree. So the element in the root has weight zero, its two children have weight 1 and so on. The you define as potential function

$$\Phi(H)=\sum_{i\in H}2  w_i.$$

Let us now analyze the heap operations. For insert you add a new element add depth $d$ at most $\log(n)$. This increases the potential by $2d$, and can be done in $O(1)$ time. Then you "bubble up" the new heap element to assure the heap-property. This takes $O(\log n)$ time and leaves $\Phi(H)$ unchanged. Thus the costs for insert are $O(\log(n)+\Delta(\Phi(H)))=O(\log n)$.

Now consider the extractMin. You take out the root and replace it by the last element in the heap. This decreases the potential by $2\log(n)$, thus you can afford to repair the heap property, and therefore the amortized costs are now $O(1)$.

If you have a general question for the potential function you should pose this as a different question.