This may be of some use to you - I have had a look at the FIAT parts
catalogue and the door hinge is similar to the Punto one which I
replaced several weeks ago.  The is not a pin as such, but a bolt that
hold to halves of the hinge together.  If this bolt works lose, it
appears that the hinge is badly worn, but put a 10 mm spanner and
tighen, and all is OK again.

If the hinge part is worn, then you need to replace it - about £15
from the dealer.  The smaller half is welded to the door, the main
part is bolted to the A frame.  Remove the single central bolt from
the top and bottom hinge and remove the pin from the middle strap
assembly.  The door should simply lift off.

Replace your hinge by removing the allen bolts but make sure you use a
marker to outline the position of the old hinge to guide you when you
tighten the new one into position.  Then, lift the door back into
position (a two man/woman job) and insert the two fixing bolts.  If
the door does not close smoothly then adjust the striker on the B
pillar.  This will work if small adjustment is only needed.

FIAT in their wisdom obviously adjust the door before fitting the
front wing - no good to the home mechanic useless you want to disturb
this panel.  It is possible to reach the hinge bolts if you remove the
protective plastic wing guards around the wheel arch, but you need
some sort of long Allen key to get at the bolts.  However, you will
not need to go to this trouble if you mark the old hinge carefully.
Believe it or not, its not as bad as it seems.