Take a differentiable manifold $M$. Define $\eta(M)$ as $\min\{\#\mathfrak{A} \mid \mathfrak{A} \text{ is an atlas for $M$}\}$. For example, if $M=S^n$, we have that $\eta(M)=2$, since $S^n$ is compact and $2$ charts (the stereographic projections) are enough to cover $M$.
Is $\eta(M)$ known for a wide range of manifolds? Is it somehow manageable to compute it? Does there exist any technique?
There is in fact a number concerned with the minimal amount of open subsets to cover $M$ which satisfy contractability in $M$ --- the Lusternik Schnirelmann category $cat(M)$. This gives you (at least with my definitiong of a chart)
$$ cat(M)\leq \eta(M). $$
There are a lot of interesting techniques presented in the literature for this, which might be interesting for you. Note that historically this category was originally defined for closed subsets, so literature can be non-consistent. Note that $cat(M)$ is also related to other fields such as Morse theory.