If your intention was for each stage $H(\alpha+1)$ to be the collection of every zero set from any set in $\bigcup_{\lambda\le\alpha} H(\lambda)$ which is a subset of $X$.
Then the answer is that $\nu(X)$ will always be 2. The reason for this is restriction preserves continuity, so (because Tietzes' extension theorem applies in this situation, and we need only consider continuous functions into $[0,1]$ and not all of $\mathbb{R}$) we have that any zero set from a subset of $X$ (which is also a zero set of $X$) is also a zero set of $X$ (this follows because we can rig up an extension which is not zero outside of the subset we are considering)
PS: If you are looking for a hard problem in this form that deals with topology, may I suggest Alan Dows' "Sequential order" chapter from "Open Problems in Topology II"