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This is my function,
(define-key global-map
(kbd "µ")
(display-message-or-buffer (message "`%s'" (eval '(solar-sunrise-sunset-string (calendar-current-date))))))
But, when I call it, through the alt+m keystroke the command literally outputs text where my cursor is, with the output.
For example, under my cursor, the text appears as so,
How can I direct this message to my Echo Area?
What you are doing is defining the key to the result of the evaluation of the expression (display-message-or-buffer (message "`%s'" (eval '(solar-sunrise-sunset-string (calendar-current-date))))). When I evaluate that expression in my *scratch* buffer, I get the string "‘Sunrise 7:04am (EST), sunset 4:10pm (EST) at 42.4N, 71.2W (9:06 hrs daylight)’". IOW, what you did is equivalent to this:
(define-key global-map (kbd "µ") "‘Sunrise 7:04am (EST), sunset 4:10pm (EST) at 42.4N, 71.2W (9:06 hrs daylight)’")
Of course, geographical differences will give you a different string, but that is irrelevant to the main point.
If you want to define the key to run a command, you first define the function that you want to run and then you give the function to define-key - like this:
(defun my/func ()
(interactive)
(display-message-or-buffer (message "`%s'" (eval '(solar-sunrise-sunset-string (calendar-current-date)))))
(define-key global-map (kbd "µ") #'my/func)
BTW, you don't need the eval: (display-message-or-buffer (message "`%s'" (solar-sunrise-sunset-string (calendar-current-date)))) will work just as well (or better: explicit use of eval is generally frowned upon - see the first paragraph of the eval section of the Emacs Lisp manual, which you can get to with C-h v i g(elisp)eval).
