Basic cons-cell sticks together two arbitrary things and is the basic unit that allows a construction of linked lists and arbitrary data objects. Question: is there a reason to stick with this simplistic language design descision (for instance, in all lisp families)?
Why not use fixed length arrays for this purpose (or some nested stacks)? I can't foresee any problems with that, but there are clear advantages of a more "packed" memory, less pointer resolution and less "dead-weight" cons-cells to define hierarchy of the data.
You have titled your question “Functional Programming: why pair as a basic constructed unit?”, but this title does not reflect correctly the fact that many important and well known functional languages (e.g. Haskell, F#, Scala, SML, Clojure etc.) have either algebraic data types or different collection of data structures, in which the pair is just one of the different type of constructors, if even available. The situation is similar for other multiparadigm languages, that have support for functional programming, like C++, Java, Objective-C, Swift, etc.
In all these cases the pair, if present, is exactly “basic” as an array, a record, or list, or any other type of data constructor.
What is left is the family of Lisp languages, notably Common Lisp and Scheme, that, beside having a rich set of data structures, like those cited in the comment of Rainer Joswig, use the pair for an important task: as basic data constructor to represent programs.
The fact that Lisp code is a s-expression (that is a list of lists and atoms) has foundamental consequences, the most notable of all being the rising of macro systems, that allow programmers to create easily new syntax, or even new domain-specific languages.