I'm a studying an intro to topology course and we have just learnt about retractions. As a side note, the lecture notes state that the ball $\mathbb{B}^{n}$ does not retract to $S^{n-1}$ in $\mathbb{R}^{n}$ (due to Brouwer).
Definition: A subset $A\subset X$ is a retract of $X$ if there is a map $r:X\to A$ such that the restriction satisfies $r|_{A}=\text{Id}_{A}$, $r$ is known is the retraction.
Suppose we had the function $$r(x) = \frac{x}{\|x\|}$$ for all $x$ except the origin, then a ball without the origin would map to the sphere. Can we not just then map the origin to some random point on the sphere? What's wrong with my understanding?
The meaning of the word "map" is context dependent.
If by "map" we understand plain function then retracts are not interesting: in this sense any subset of any set is a retract of that set. And your solution is one possibility.
But in the context you are dealing with, the word "map" means "continuous function" (and we know, or at least assume that, because you've mentioned Brouwer, spheres, balls, i.e. topological context). And note that your newly constructed function is not continuous, regardless of the choice of $r(0)$. Why? Hint: take any point $v\in S^{n-1}$, $v\neq r(0)$ and consider the sequence $r\big(\frac{1}{n}v\big)$.
Also there are other interesting worlds with retracts, e.g. algebraic structures and homomorphisms. In fact the concept of "retraction" can be defined for any category.