Assume events occur as a Poission process with parameter $\lambda$ in a fixed size interval.
Assume $Y$ is a random variable, $Y = \frac{\text{count of intervals with 0 events}}{\text{total count of intervals}}$
Therefore we know that
$E(Y) = P(0 \text{ events occur})$
or $E(Y) = e^{-\lambda}$.
Now I observe a value of $Y$ from a single data point as $Y=y$. Can I estimate $\lambda$ as a function of $y$?
Note that -- conditional on the total number of fixed-size intervals ($n$) -- the number of them with counts of 0 will be $\textit{binomial}(n,e^{-\lambda})$.
In that case, this looks like a straightforward inference problem for which a simple maximum likelihood (ML) estimator for $e^{-\lambda}$ exists, and hence one for $\lambda$ is available (though you need the number of counts of 0 - let's call that $N_0$ - to be non-zero for it to make any sense, and even conditioning on $N_0>0$, it will be biased).
You might consider if you can find ways to modify it (/regularize it) to make a more suitable estimator. For example, it might help to get started along those lines if you consider ways (other than ML) to estimate the binomial parameter when the sample proportion is 0 (the ML estimate is 0, but given that the parameter is in (0,1) that's not much use).
