This is my attempt at the question.

Assume f's Laurent series is not unique.

Then, we have

$$f(z) = \sum a_{n} (z-z_{0})^n$$

$$f(z) = \sum b_{n} (z-z_{0})^n$$

Subtract the two equations and get

$$0 = \sum (a_{n}-b_{n}) (z-z_{0})^n$$

So, we must have

$$a_{n} = b_{n}$$ and f's Laurent series must be unique.