In
Ajtai and Ben-Or. A theorem on probabilistic constant depth Computations. STOC '84, 1984
Ajtai and Ben-Or show a non-uniform derandomization of BPAC0.
Is there a similar relation known for probabilistic and non-uniform NC0? Or is it an open problem?
Any $NC^0$ circuit family $(C_n)_{n \geq 0}$ can only depend on a constant number of bits, whether they're input bits or random bits. Let's say there are $n$ inputs and $r$ random bits, but our circuit $C_n$ only depends on $n_0$ inputs and $r_0$ random bits, which are both fixed independent of $n$. So, create $2^{r_0}$ copies of $C_n$ and hardwire each possible combination of the $r_0$ random bits into each copy of the circuit. Note that $2^{r_0}$ is still a constant, so we can take the majority of these copies of the circuit and get our answer deterministically. The depth of such a circuit will be the depth of $C_n$ plus the depth of the MAJ circuit, which is still a constant, so the resulting deterministic computation is still in $NC^0$.
For an example of a constant-depth circuit class that isn't known to be closed under randomness, look at $CC^0$, the class of unbounded-fanin polynomially-sized constant-depth circuits of $MOD_m$ gates for fixed $m$. It is known that $BPCC^0 = BPACC^0 = ACC^0$ (see Hansen and Koucký below), so derandomizing $CC^0$ is equivalent to showing $CC^0 = ACC^0$.
Hansen, Kristoffer Arnsfelt; Koucký, Michal, A new characterization of (\text{ACC}^{0}) and probabilistic (\text{CC}^{0}), Comput. Complexity 19, No. 2, 211-234 (2010). ZBL1213.68262.
