Can we define a height function for a variety over a finite field? That is, is there a way to measure the complexity of a point over a finite field the same way we do it over number fields?
 A: What properties do you want your "height" to have? A possibly better approach is to figure out those desired properties and then see if you construct such a function (or prove it can't exist, in which case you'd need to lower your expectations). For example, a "good" height function on the $\mathbb Q$-rational points of varieties should have two properties: (1) it measures complexity, so in particular there are only finitely many points of bounded complexity and (2) it is reasonably functorial for the maps that one studies in algebraic geometry, so for morphisms and/or rational maps. For a logarithmic height function, a natural complexity measure is (roughly) the number of bits it takes to store the point (or whatever object you're studying) on a computer. Of course, one may later want to refine this, but it's a good starting point. Then for morphisms $f:\mathbb P^n\to\mathbb P^m$ of projective space, we get the funtoriality in the sense that $h(f(P))=(\deg f)h(P)+O_f(1)$ for all $P\in\mathbb P^n(\mathbb Q)$. However, if $f$ is a rational map, we only get an upper bound
$h(f(P))\le(\deg f)h(P)+O_f(1)$.
Now let's consider $\mathbb P^n(\mathbb F_p)$, or maybe $\mathbb P^n(\overline{\mathbb F}_p)$. Will's suggestion is $$h(P)=\log\min_{P=[a,b]}\{|a|,|b|\}.$$ That's a good complexity measure, i.e, it's fine for (1), but it has no apparent functoriality. One of Dror's suggestions is $h(P)=[\mathbb F_p(P):\mathbb F_p]$. This is okay for (1), also, at least if we think of $p$ as fixed, since every element of $\mathbb P^n(\mathbb F_{p^k})$ can be specified by $O(k)$ bits, where the $O$ constant depends on $n$ and $p$, but not on $k$. And now we get a sort of subadditive functoriality, since $h(f(P))\le h(P)$. But that's actually not that good, we really want $f(P)$ to be more complicated than $P$, at least if $f$ is "complicated" and $P$ is not "special".
So this isn't really an answer to your question, but maybe will help you to think about your question in a different way.
