Say you have a standard GBM for the underlying asset price for Black-Scholes, written as a martingale:
$d\hat{S}/\hat{S} = \sigma d\hat{B}(t)$.
If you wanted an expression for $\hat{S}(0)$, is this just by Martingales:
$\hat{S}(0) = \mathbb{E}[\hat{S}(T) | \hat{S}(0)]$ ?
This is strange to me, because it is saying to find $\hat{S}(0)$, you have to know what $\hat{S}(0)$ is, as your expectation is conditioned on it, or is my interpretation incorrect?
Your formula is correct (as you say, it's just an expression of the martingale property), but you seem to be having some confusion on its interpretation. Usually we would write it the other way: $$ E[S(T)\mid S(0)] = S(0),$$ in other words, we're computing the LHS based on the RHS. This is because $S(0)$ usually represents the stock price right now, which is a known quantity. The left hand side is the expected value of the stock at time $T$ in the future given that we know the stock price now. The equation says that the average price at time $T$ is just the price right now.
So yes when you write an equation like that, since the LHS is conditioned on $S(0)$ you're generally assuming you know the value of $S(0).$ Actually, that's not strictly true, though. If we were sitting at time $-1$ or something and didn't know yet what $S(0)$ was, the equation $E[S(T)\mid S(0)] =S(0)$ would still be true and make sense. What it says is, roughtly, "once I know the value of $S(0)$ (which is a random variable) the value it takes will be the expected value of the price at time $T$."
Note that still this is not a way to "calculate" $S(0)$ (it is random). What you can calculate, though, is its expected value given what you know now: $E[S(0)\mid S(-1)] = S(-1)$