To me, it seems that the operation mirror to the changing of the base field is the changing of coefficients in Floer homology. Let me give you some examples.
For the case when $k$ is any field, we have the following example: Take $\mathbf{P}^2_k$ as our variety and its mirror $W: \left(\mathbf{C}^{\times}\right)^2 \rightarrow \mathbf{C}$, $W(x,y) = 1 + x + y - 1/xy$. For the Fukaya-Seidel category of vanishing cycles of $W$ take the coefficients in $k$ and forget about weighting by exponentials of the areas of holomorphic polygons. Then, the bounded derived category of coherent sheaves on $\mathbf{P}^2_k$ is equivalent the idempotent-completed derived Fukaya-Seidel category of $W$ with coefficients in $k$. In fact, the first statement of this result in writing, in Seidel's More on vanishing cycles and mutation, sets $k = \mathbf{Z}/2\mathbf{Z}$.
For a quartic surface, we know one side of mirror symmetry holds when $k$ is the rational Novikov field over $\mathbf{C}$, $\Lambda_{\mathbf{Q}}$. Precisely, we have an equivalence between the idempotent-completed derived Fukaya category of a smooth quartic surface over $\mathbf{C}$, with coefficients in $\Lambda_{\mathbf{Q}}$, and the bounded derived category of the mirror of a smooth quartic surface over $\Lambda_{\mathbf{Q}}$. Here it seems perfectly plausible to replace $\mathbf{C}$ by $k$ again. However, there is a significant difference with the previous example. For $\mathbf{P}^2$, we never had to worry about convergence of the power series defining the products in the Fukaya category thanks to the exactness of everything in sight. But, here a lot of important questions are over $\mathbf{C}$ and depend on convergence. So it would make the most sense to take something like a $p$-adic field for $k$.
Changing coefficients may not seem very sexy and it probably will not have much to say about GW invariants of varieties over finite fields, but it may nonetheless provide interesting results. The first case to investigate: try mirror symmetry for an elliptic curve over a $p$-adic field. As first step, can one reproduce a statement like that of Polishchuk and Zaslow?
Caveat emptor: I have no idea, but I think it would be interesting to find out.