Question: I would like to have a link that runs a Babel source-code block. Does Org-mode support running something from the library of babel in a link, or a source block via a file+blockname reference? A Hyperbole button would also be acceptable.

Answer: 
You can easily define your own link type that runs source blocks:

```el
(org-link-set-parameters
 "run"
 :follow #'org-babel-ref-resolve)

```

Afterwards a click on `run:hello` executes the source block in the following Orgmode file.

```
#+name: hello
#+begin_src emacs-lisp :var n=1
(message "Hello. n=%s" n)
#+end_src

#+RESULTS: hello
: Hello

run:hello

[[run:hello(n=2)]]

```
The usual way to do this is like this:

```
#+name: mysum
#+BEGIN_SRC emacs-lisp
(+ 5 5)
#+END_SRC

#+call: mysum()

#+RESULTS:
: 10

```

I am not sure what the point of doing this in a link would be, where would the results go for example?

You can make a link like this: 
[[elisp:(progn (org-babel-goto-named-src-block "mysum") (org-babel-execute-src-block-maybe))]]

which will jump to that block and run it.