$BPL$ with polylog random bits is in $L$
It follows from this PRG of Nisan and Zuckerman. This paper shows that if you have an algorithm that uses space $S$ and only $\mathrm{poly}(S)$ random bits, then the number of random bits can be decreased to $O(S)$.

In particular, in the setting you describe, we have $S = O(\log n)$, so the number of random bits can be reduced to $O(\log n)$. Then, we can derandomize the algorithm completely by enumerating over all possible coin tosses.