According to Dr Sharib:

"Al-Hakam is the One Whose order or judgement cannot be annulled or
set aside. His judgement is ultimate and His order is final. He decides
what is good and bad, and what is true or false...".
Metaphorically this name applies to a person who decides between two

To emulate the name we are advised to decide which of our desires (or
course of action) would meet with Allah's approval and to stick to that -
renouncing any other. We should dispense justice when called upon to
do so without feeling proud of the power confered upon us to do so. We
should be aware that such power could be removed by He Who gave it.

The reward of appropriate invocation is that the heart becomes a
repository of the secret of Allah and of being the recipient of intuitions
and inspirations.

see 'The 99 Most Beautiful Names of Allah' by Dr Zahurul Hassan Sharib

Some other references:

l-Ghazali gives an example of the Just ruling of God:-

'Man has only what he strives for and that his effort will be seen.' (Q.

The religious profit to be gained from beholding this attribute of God
Most High is to know that the matter is settled and not to be appealed.
For the pen is already dry, (having written) what exists. The causes are
already applied to their effects, and their being impelled to their effects
in their proper times is a necessary inevitability.

(Al-Ghazali - The Ninety-nine Beautiful Names of God. Trans. Burrel/Daher: ITS: 1992.)

As far as I have been able to understand it this attribute is associated
with the decisive action of God in bringing into the unseen world some
new situation already complete in its generality and its particulars. Its
coming into being in the material universe follows inevitably as a
reflection of this. It is assumed by those who are the unknowing tools of
this 'bringing into being', that they are causal agents but it is thus the fate
of nations and individuals, the development of society and of new
inventions is brought about.

Al-Ghazali points out that examples to explain this have their limitations
and dangers, but one could thinks of it as a mold whose shape has been
decisively set. During the pouring into the mold of the individual particles
of the melted iron or wax they will will run according to the prescribed
nature - one will run this way one another but the shape of the mold alone
will dictate the shape they finally become

So far as I have been able to understand it, the quality of God known as
Al-Hakam is that which initiates decisive, irrevocable action. The
comparison in modern human experience is therefore not only that of the
magistrate or judge but that of the good manager or leader who, having
listened to all the arguments of his team, all the feedback from the
experts and pressure groups, having considered the moral and ethical
implications, etc. and weighed it all up, decides a particular course of
action and initiates
it putting aside all alternatives.

Bismillah ir Rehman ir Rahim
(Qur'an 40:48)
The Judge
The Arbitrator, The Arbiter
Those who had been arrogant
Will say, "We are all
In this (Fire)! Truly,
Allah has judged
Between His servants.
'innallaaha qad hakama

(Qur'an 40:48 trans. Y. Ali)