Short version: One can define a version of the Lefschetz fixed point theorem using any homology or cohomology theory. All versions will be true on some topological spaces, since they agree on some topological spaces, but some might be true more generally than others. If two versions have the same generality, one might be more exact, and make stronger statements. Under this criterion of goodness, which one is best?
Potential answer: Is the the theorem true for Cech cohomology on all T3 compact spaces?
Intuition: Suppose there is a continuous map from a topological space X to itself. Does this have a fixed point?
This is an extraordinarily elementary question. It makes sense to ask this question about any topological space. Yet the tools that are typically used to answer it are usually very specific, referring to special topological spaces like $\mathbb R^n$.
The Lefschetz fixed point theorem is a powerful result in this area. It is usually defined in terms of singular homology, which is not an elementary construction at all. But one could easily use another homology or cohomology theory to calculate the matrix chases and, thereby, the number of fixed points.
In particular Cech cohomology seems extraordinarily elementary, being defined entirely in terms of open sets and their intersections. Therefore, one would expect Cech cohomology to be the best one.
Consider, as an example, the topologist's sine curve. All maps from it to itself have a fixed point. But under singular homology it has characteristic 2, not 1. Cech gives characteristic 1 and is thus more exact.
Edit: I'm more interested in the weak than the strong version of the theorem, because the weak version is presumably more generalizable. I'm interested in any sort of information that compares the theorem in different theories even if it doesn't fall into a strict/better worse dichotomy.