The lore of the game Numenera mentions "an irrational number that may be a four-dimensional equivalent of $\pi$". What could this mean?

There's an RPG (Role-Playing Game) called Numenera, set on the Earth a billion years in the future, which is covered in the partially-functional and generally weird and mysterious technological ruins of a billion years of advanced civilizations. In one of its sourcebooks, it lists possible fragmentary transmissions that a player character might receive from its global "datasphere", one of which is listed as the following:

An irrational number that may be a four-dimensional equivalent of $\pi$.

When I saw this, my first thought was "I'm pretty sure that's probably just pi multiplied by a constant", followed by "If it's not, I'm sure mathematicians have already worked it out." When I did I Google search, I couldn't find anything obvious.

So, what is the equivalent to pi for a four-dimensional hypersphere?

You're close. The "volume" of a $4$-dimensional ball is given by $$ V = \frac{\pi^2}{2}R^4 $$ and its "surface area" is given by $$ S = 2 \pi^2 R^3. $$ If we take the $n$-dimensional equivalent of $\pi$ to be the ratio between the volume of the $n$-ball and $R^n$ (the volume the $n$-cube with side length $R$), then the 4-D equivalent of $\pi$ is $\frac{\pi^2}{2}$.

More generally, we would have (for positive integers $k$) $$ \begin{align}\pi_{2k} &= \frac{\pi^k}{k!}, \\ \pi_{2k+1} &= \frac{2^{k+1}\pi^k}{(2k+1)!!} = \frac{2(k!)(4\pi)^k}{(2k+1)!}.\end{align} $$ where $\pi_n$ is the $n$-dimensional equivalent of $\pi$. Because $\pi$ is known to be transcendental, we can conclude that these are all irrational (and transcendental as well).

If you mean the sphere in the 4-dimensional Euclidean space, then the ratio between its surface area and its radius cubed is $2\pi^2$