In elisp, how do I put a function in a variable?

Note that in Emacs Lisp, symbols have a value cell and a function cell, which are distinct. When you evaluate a symbol, you get its value. When you evaluate a list beginning with that symbol, you call its function. This is why you can have a variable and a function with the same name.

Most kinds of assignment will set the value (e.g. let, setq, defvar, defcustom, etc...) -- and as ryuslash shows you can assign a function as a value and call it via funcall -- but there are also ways to assign to a symbol's function cell directly using fset (or flet, defalias, etc)

(fset 'my-function 'dumb-f)
(my-function)

In your case I would use ryuslash's answer (except you need to use defcustom rather than defvar if you want it to be available via customize).

Also, regarding "I'm having a hard time googling for it"; always remember that Emacs is self-documenting, and among other things contains a good manual complete with index (well, more than one manual, in fact). So even if Google remains your first port of call, it shouldn't also be your last if you can't find what you're looking for. From the contents page of the elisp manual you can navigate to "Functions" and then "Calling functions" and you will be told about funcall almost immediately.

To run a function stored in a variable you can use funcall

(defun dumb-f ()
  (message "I'm a function"))

(defvar my-function 'dumb-f)

(funcall my-function)
==> "I'm a function"