Let $(A, \leq)$ be a poset. A set $B \subseteq A$ is said to be cofinal in $A$ if every $a \in A$ has a $b\in B$ with $a \leq b$.
Why is a cofinal set called "cofinal?" Is there a related notion of a "final" set? The "co" doesn't seem to be related to the word "complement," as the complement of a cofinal set can also be cofinal; such sets don't seem to be "final" in any natural sense of the word I can imagine.
On
The term cofinal is a compound of the prefix co- and the adjective final. Here the meaning of the prefix is ‘together; mutually; jointly’: if $B$ is cofinal in $A$, the two sets reach the ‘end’ together, so to speak. The prefix is from Latin co-, a variant of con-, and indeed one occasionally sees confinal in place of cofinal. The Latin prefix con- in turn is derived from the preposition cum ‘with, along with’.
The OED does not yet have entries for either cofinal or confinal, so I can’t say when either came into use in the mathematical sense.
I'm not sure who came up with the term. For me it suggest that the set is "coming with you to the end", co- as in compansion, co-traveller; -final for end, the larger elements are towards the end in an order, intuitively. So whatever direction you go towards larger elements, there will always be elements from this cofinal set too..
But one should find the first usage of the term and maybe it's explained there? Not sure if Cantor used e.g.