Although the question is phrased a bit sloppily, there is a standard interpretation: Given a smooth complex proper variety $X$, choose a smooth proper model over a finitely generated ring $R$. Then one can reduce modulo maximal ideals of $R$ to get a variety $X_m$ over a finite field, and ask what information about $X$ it retains. As has been remarked, the zeta function of $X_m$ gives back the Betti numbers of $X$.
I believe Batyrev shows in this paper that the zeta function of a Calabi-Yau is a birational invariant and deduces from this the birational invariance of Betti numbers for Calabi-Yau's. And then, Tetsushi Ito showed here that knowledge of the zeta function at all but finitely many primes contains info about the Hodge numbers. (He did this for smooth proper varieties over a number field, but a formulation in the 'general' situation should be possible.)
For an algebraic surface, once you have the Hodge numbers, you can get the Chern numbers back by combining the fact that
$c_2=\chi_{top},$
the topological Euler characteristic, and Noether's formula:
$\chi(O_X)=(c_1^2+c_2)/12.$
I guess this formula also shows that if you know a priori that $m$ is a maximal ideal of ordinary reduction for both $H^1$ and $H^2$ of the surface, then you can recover the Chern numbers from the zeta functions, since $H^1(O_X)$ and $H^2(O_X)$ can then be read off from the number of Frobenius eigenvalues of slope 0 and of weights one and two.
You might be amused to know that the homeomorphism class of a simply-connected smooth projective surface can be recovered from the isomorphism class of $X_m$. (One needs to formulate this statement also a bit more carefully, but in an obvious way.) However, not from the zeta function. If you compare $P^1\times P^1$ and $P^2$ blown up at one point, the zeta functions are the same but even the rational homotopy types are different, as can be seen from the cup product in rational cohomology. See this paper.
Added: Although people can see from the paper, I should have mentioned that Ito even deduces the birational invariance of the Hodge (and hence Betti) numbers for smooth minimal projective varieties, that is, varieties whose canonical classes are nef. Regarding the last example, I might also point out that this is a situation where the real homotopy types are the same.
Added again: I'm sorry to return repeatedly to this question, but someone reminded me that Ito in fact does not need the zeta function at 'all but finitely many primes.' He only needs, in fact, the number of points in the residue field itself, not in any extension.
This doesn't quite sound like the right conjecture here, because Z is known to go to infinity on the average, by Selberg's central limit theorem (see my blog post on this topic). But this is easy to fix by working with a projective notion of local quasiperiodicity in which one divides $f(t)$ or $f(t+t_n)$ by an $n$-dependent scaling factor. In that case, one is basically asking for the zero process of the zeta function to be recurrent, and this would be predicted by the GUE hypothesis. However, I doubt that this question will be resolved before the GUE hypothesis itself is settled.
EDIT: Note though that there are other hypotheses than the GUE hypothesis that also lead to a recurrent zero process, such as the Alternative hypothesis, which is linked to the existence of infinitely many Siegel zeroes. I suppose it is a priori conceivable that some sort of dichotomy might be set up in which recurrence is obtained by completely different means in each case of the dichotomy (as is the case with proofs of multiple recurrence in ergodic theory) but I am personally skeptical that one could really handle all the cases without making enough progress on understanding zeta to solve much more difficult and prominent conjectures about that function. (In particular, with this approach one would have to first eliminate the possibility of having only finitely many zeroes off the critical line, leading us back to the original conjecture that motivated the one here.)
Best Answer
This history is described in Euler and the Zeta Function by Raymond Ayoub (1974). In his early twenties, around 1730, Euler considered the celebrated problem to calculate the sum $$\zeta(2)=\sum_{n=1}^\infty \frac{1}{n^2}.$$ This problem goes back to 1650, it was posed by Pietro Mengoli and John Wallis computed the sum to three decimal places. Ayoub conjectures that it was Daniel Bernoulli who drew the attention of Euler to this challenging problem. (Both lived in St. Petersburg around 1730.)
Euler first publishes several methods to compute the sum to high accuracy, arriving at $$\zeta(2)=1.64493406684822643,$$ and finally obtained $\pi^2/6$ in 1734. (We know this date from correspondence with Bernoulli.) It was published in 1735 in De summis serierum reciprocarum.
$1+\frac{1}{4}+\frac{1}{9}+\frac{1}{16}+\frac{1}{25}+\frac{1}{36}+\text{etc.}=\frac{p^2}{6}$, thus the sum of this series multiplied by 6 equals the square of the circumference of a circle that has diameter 1. [Notice that the symbol $\pi$ was not yet in use.]
The generalization to $\zeta(s)$ with integers $s$ larger than two followed in "De seribus quibusdam considerationes". In 1748, finally, Euler derives a functional equation relating the values at $s$ and $1-s$ and conjectures that it holds for any real $s$. (Euler's functional equation is equivalent to the one proven a century later by Riemann.).