In Computer Science, the word "Map" often refers to an associative array, i.e. a list of key-value pairs. You could think of this sort of data structure as being like a dictionary.
Likewise, as a verb, "mapping" refers to the process of associating keys with values.
Based on some dictionary searching, it seems that "mapping" as a verb has long meant to make a map, i.e. to draw out a representation of a geographic space.
I also see references to use in Mathematics that predate the use in Computer Science, which makes sense. OED provides this example of mathematical use:
1939 M. H. A. Newman Elem. Topol. Plane Sets of Points i. 12 A (1, 1)-correspondence is set up between the set of all positive integers, I, and the set of positive even integers, E, by mapping n of I on 2n of E.
However, it seems like using "map" as a verb meaning to associate values with each other is a major shift from the original meaning related to a graphical representation of geography.
So how did "map" evolve to have such a specific technical definition that seems quite different from its original meaning?