What does "map" mean?
So, there are two distinct uses of the word "map", that I'll unpack here.


The first is very generic, where map means "to associate," particularly by way of a function. If we say "$f$ maps each $x$ to $2x$", then we're saying $\forall x \ldotp f(x) = 2x$.

This usage includes "memory mapped IO:" there is a (conceptual) function associating each piece of memory with a particular IO action. Nobody actually ever writes out the function, but it is indeed there: for each piece of mapped memory, there's some IO associated with it. Maybe a part of a disk, maybe a hardware register on a peripheral, etc.

Likewise, bit arrays (and arrays in general) fall into this: each index has a single element associated with it (at any given time), so an array is effectively an encoding of a finite-domain function.
In functional programming and derivatives (such as MapReduce), map refers to applying a transformation across a structure.

The original map comes from Lisp, where it referred to the function that took another function and a list, and returned the result of applying the function to each element of that list.

But, this phenomenon is quite general. In Haskell, a data structure that admits such an operation is called a functor, and the operation is called fmap (for historical reasons, to avoid conflicts with the list map).

These all are related through the concept of a Functor from category theory, which is an abstraction of structures admitting a "map" operation.