Was this complexity measure studied before? An initially unknown string $x$ is known to be in $C\subseteq \{0,1\}^n$. Every round you are allowed to reveal one bit of $x$ (on your choice, and adaptively). How many bits do you need to reveal in order to be certain about what $x$ is?

For example, if $C=\{x\mid \text{parity}(x)=0\}$, then $(n-1)$ bits are needed. If $C=\{e_i\mid i\in [n], e_i=(0,0,\cdots,0,1(\text{i}^{th}\text{coordinate}),0,\cdots, 0)\}$ then also $(n-1)$ bits are needed. But if $C=\{0^p1^{n-p}\mid p\in [n]\}$ then only $\log n$ bits are needed since we can binary search the position where the alternation happens.

Was this "revealing complexity" $RC(C)$ studied before? If not, what is some combinatorial feature (or any sufficient/necessary conditions) of $C$ with $RC$ to be $n-o(\log n)$?

Any previous results or new ideas are appreciated.
What you call the "revealing complexity" is sometimes called the (deterministic) query complexity of exact learning with membership queries. 

Exact learning refers to the fact that you have to exactly identify $x\in C$ at the end, as opposed to being (say) probably approximately correct (as in PAC learning). Membership queries refers to the fact that you can query the individual bits of $x$ (as opposed to being given a random sample according to some distribution, as in PAC learning). 

See this related question where I explain some of what's known about this complexity measure.
For example, we know that the deterministic query complexity of exactly learning a concept class $C\subseteq \{0,1\}^n$ with membership queries is characterized up to a factor of $\log |C|$ by the complexity measure defined in this paper by Servedio and Gortler or the extended teaching dimension of Hegedus. Both measures are combinatorial in nature. Whether there is a combinatorial measure that captures this complexity without a $\log|C|$ factor is unknown to me (and probably an open problem).