Thank you for your replies, I think i'm beginning to understand the
process. These are the changes I made to local.mk:

prefix  = /usr/local/share
datadir = /usr/local/share/emacs/24.4.50/etc/org (actually, this directory
belongs to the built-in Org installation)

Then:
$ make
$ sudo make install

And it worked fine. The info manual is now updated, so is everything else.

So, overall, the purpose of running 'make install' (or 'make up2', to also
pull from the Git repository) is to avoid overriding the default Org
installation that comes with Emacs?

Going through the variables in local.mk it seems this is the idea, except
maybe for 'infodir' which in my case points to the place where Emacs threw
all its info files (including Org).

To sum up, the process made the following changes on my system:
- Org lisp files from Git now reside in /usr/local/share/emacs/site-lisp
(which is in load-path by default). The original built-in Org lisp files
are untouched.
- M-x org-version shows the correct updated version and location.
- M-x info shows the updated info manual which now resides in
/usr/local/share/info (I think it replaced the built-in manual?).
- Org data files now reside in /usr/local/share/emacs/24.4.50/etc/org
(replacing the built-in files?).

No extra elisp code was needed (adding the cloned repository to load-path,
etc.).