According to Dr Zahurul Hassan Sharib:

"Al-Muhaymin, according to some people, is He Who is the Best, the Noblest,
and the Highest. According to others Al-Muhaymin is the One Who is
responsible for the sustenance of the people."

'To emulate this name you should protect yourselves from inordinate desires.
You should try to overcome your self, to take to introspection and realise and
feel that nothing is hidden from Allah.'

Appropriate recitation of this name is conducive to purification and illumination
of the inner and outer life.

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

Some other references:

Al-Ghazali refers to the aspect of guardianship that implies taking command of
a situation (from knowledge), possessing it (from perfection of power) , and the
action of guarding it.

He says: "Every servant who watches over his heart until he supervises its
depths and its secrets, and also takes possession of reforming his inner states
and attributes, and undertakes to protect it continuously, according to the
requirements of his reform, will then be guardian in relation to his heart."
According to him those who knowing the inner state of others undertakes to
keep them on the right way have an even greater share in this quality.

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

Shaykh Tosun Bayrak says: 'He is the one who sees to the evolution and
growth of His creation ......nothing escapes His attention.'

The Most Beautiful Names: Sheikh Tosun Bayrak al-Jerahi al-Halveti:Threshold Books 1985

Sir Edwin Arnold illustrates the name with the story of the miraculous escape of
the holy Prophet and his companions from his pursuers on account of the
spider's web that fooled the enemy into thinking that none could have recently
entered that cave. His verse runs thus:

Thus did a desert bird and spider guard
The blessed Prophet then;
For all things serve their Maker and their God
Better than thankless men

Allah-al-Muhaiman! Shield and save
Us, for his sake within that cave.

Sir Edwin Arnold: Pearls of Faith.
Bismillah ir Rehman ir Rahim
The Protector, The Vigilant, The Guardian,
The Giver of Safety, He Who is the Best,
the Noblest, and the Highest
Qur'an 59:23
Allah is He, than Whom
There is no other god:-
The Sovereign, the Holy One,
The Source of Peace (and Perfection),
The Guardian of Faith,
The Preserver of Safety,
The Exalted in Might,
The Irresistible, the Supreme.
Glory to Allah!
(High is He)
Above the partners
They attribute to Him.
Laaa-'illaaha 'illaa Huu; -
'Al-Malikul - Qudduusus
- Salaamal
- 'Mu'minul
- Muhay-minul
- Aziizul
- Jabbaarul - Mutakabbir:
Subhaanallaahi 'ammaa

(Qur'an 59:23 trans Y. Ali)