Name for the set of possible functions $A\to B$

Another piece of notation for this is $B^A$. The reason is because the cardinality of this set is the same as $|B|^{|A|}$.

$B^A$ is common in mathematics.

Computer scientists (or at least the undersigned computer scientist) will sometimes prefer to cut the notational crap and declare "$A\to B$" to be the name of the set of maps from $A$ to $B$.

Some typical notation would be $\mathrm{Hom}(A,B)$.This is the collection of maps from $(A,B)$, and in $\mathrm{Set}$, these are just usual functions.

However, this may just be habit for me, I think Alfred Yerger's notation is far more standard for sets.

The name is exponentiation. See e.g. Smith, Romanowska - Post-Modern Algebra