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

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 $2\pi^2$

Another possible generalization, that extends the "circumference to diameter" concept more directly, is to consider the ratio of the surface volume to the volume of a diametric cross-section, the latter of which is a sphere. In three dimensions, the equivalent is the ratio of surface area of the sphere to the area of a planar cross-section taken through the center (which is a circle), and to see the relationship to the two-dimensional case, note that "circumference" can be thought of as the "surface length" and "diameter" as the length of a linear cross-section through the center.

That is, in effect, the "$\pi$" is the ratio of the surface volume of a 4D ball of radius $R$ to the regular volume of a 3D ball of radius $R$. Thus...

The surface volume of the 4-dimensional ball is

$$\mathrm{SV} = 2\pi^2 R^3$$

and the volume of the 3-dimensional ball is

$$V = \frac{4}{3} \pi R^3$$

so the ratio is just

$$\pi_4 = \frac{\mathrm{SV}}{V} = \frac{2\pi^2 R^3}{\frac{4}{3}\pi R^3} = \frac{3}{2} \pi$$

Hence, "four-dimensional pi" is $\frac{3}{2} \pi$. You're welcome. And yes, it is $\pi$ times a constant (rational number, as @rghome mentions in the comments).