A very useful invariant associated to any group is the smallest possible cardinality of a generating set that generates that group. This is always guaranteed to exist, no matter what the group, at least given the axiom of choice. To see this, we can simply look at the set of all generating sets for that group, then map each element in that set to the cardinal it's in bijection with. Then, we have a set of cardinals, and at least using the axiom of choice we know this must have a least element because cardinals are well-ordered.
For a finitely-generated free abelian group it's equal to the rank of the group, but in general it's different. For instance, the group $\Bbb Z/2\Bbb Z \times \Bbb Z/3\Bbb Z$ has rank 0, but the smallest generating set has cardinality 1. It also is well-defined even for non-finitely-generated groups: the group $\Bbb Q$ isn't finitely generated, and the smallest generating set has cardinality $\aleph_0$. And so on.
Does this quantity have a name or play any kind of important role in the usual approach to group theory?
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To convert my comment to an answer:
The (relatively) standard terminology is that the minimal cardinality of a generating set of a group $G$ is called the rank of $G$. It is primarily used in the case of finitely generated groups, see here. One of the earlier uses of this notion was in the proof of Grushko's theorem: The free product decomposition of a finitely generated group eventually terminates. Grushko proved this by establishing the equality: $$ rank(G_1*G_2)= rank(G_1) + rank(G_2), $$ provided that $G=G_1*G_2$ is finitely generated.
It is an unfortunate fact that the word "rank" has other meanings in group theory as well: In the theory of algebraic groups (and Lie groups) it means something completely different (for semisimple algebraic groups, this is the dimension of a maximal split torus) and, as you know, it has a different meaning for abelian groups as well.