The distributional derivative of any increasing function is a positive measure on the real line. The Wikipedia article hints at what this measure is for the Cantor function $f$:
[the Cantor function] is the cumulative probability distribution function of a random variable that is uniformly distributed on the Cantor set. This distribution, called the Cantor distribution, has no discrete part. That is, the corresponding measure is atomless.
The measure $f'$ can be described as the pushforward of the Lebesgue measure under the map described in terms of binary and ternary expansions as: $$0.01100010111_2\mapsto 0.02200020222_3$$
Alternatively, the measure can be obtained as the limit of the natural measures on pre-Cantor sets: just restrict the Lebesgue measure to pre-Cantor set $C_n$ and normalize it so that the total measure is $1$. Let's call the resulting measure $\mu_n$. The weak* limit of $\mu_n$ exists and is the distributional derivative of the Cantor function.
And here is why. Let $f_n$ be the cumulative distribution function of $\mu_n$. Then $f_n$ is continuous and piecewise linear; in particular it's a Lipschitz function. Hence, its distributional derivative is its pointwise derivative, which is [the density function of] $\mu_n$. Observe that $f_n$ converge uniformly to the Cantor function $f$. Uniform convergence implies convergence in the sense of distributions. And when $f_n\to f$ distributionally, it follows that $f_n'\to f'$ distributionally.