Arabic Root:
From the root ha-mim-nun (ه م ن), which has the following classical Arabic connotations: to watch over, oversee, protect, guard to be a witness to offer security and peace to determine what is true, to extend a wing. Similar to the Arabic expression, 'haymana al-ta'ir alafirdkhih' meaning that the bird took its young ones under the protection of its wings.

The Absolute Authority of Al-Muhaymin:
The name Al-Muhaymin refers to His supreme nature as being the guardian or the absolute authority. He is the superseder, nothing will overtake or usurp Him. He is the ever-watchful, not just limited to the superficial, He can see all which is concealed in the hearts. Nothing can be kept a secret from Him. "He knows what is in the heavens and earth; He knows what you conceal and what you reveal; God knows very well the secrets of every heart." (64:4)

You can find variations of the word muhaymin, like muhaiminan in Surah Maidah verse 48 where Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ declares the Qur'an as being the book that will be well-guarded over the other books:

Muhsin Khan Translation:
"And We have sent down to you (O Muhammad ﷺ) the Book (this Quran) in truth, confirming the Scripture that came before it and Mohayminan (trustworthy in highness and a witness) over it (old Scriptures)."

But as a name and attribute of Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ, Al-Muhaymin is only ever mentioned once in the Qur'an.

هُوَ ٱللَّهُ ٱلَّذِي لَآ إِلَٰهَ إِلَّا هُوَ ٱلۡمَلِكُ ٱلۡقُدُّوسُ ٱلسَّلَٰمُ ٱلۡمُؤۡمِنُ ٱلۡمُهَيۡمِنُ ٱلۡعَزِيزُ ٱلۡجَبَّارُ ٱلۡمُتَكَبِّرُۚ سُبۡحَٰنَ ٱللَّهِ عَمَّا يُشۡرِكُونَ

Huwal-laahul-lazee laaa Ilaaha illaa Huwal-Malikul Quddoosus-Salaamul Muminul Muhaiminul-aAzeezul Jabbaarul-Mutakabbir; Subhaanal laahi Ammaa yushrikoon

English Translation:
"He is Allah, other than whom there is no deity, the Sovereign, the Pure, the Perfection, the Bestower of Faith, the Overseer, the Exalted in Might, the Compeller, the Superior. Exalted is Allah above whatever they associate with Him."— (Qur'an 59:23)

A sense of ownership/authority:
The word هيمنة (transliteration: haymana) in modern Arabic means dominance. One country can seize another, being in control or showing authority over that state. But this is often transitory, exchanging hands from one "owner" to another. Only Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ is Al-Muhaymin, the one who has complete guardianship, a quality of being in control, supremely that lasts for eternity. It is not limited by any means.

But this is one form of dominance. Muhaymin also refers to an authoritative figure who reigns supreme, one that you can trust. This is analogous to a parent who out of love for their child will set rules and boundaries to protect and care for the child. It's often done in the best interest despite the kid thinking otherwise. Imam Al-Ghazali says, "Everyone who has complete command of a situation, who takes possession of it and protects it, will be its 'Guardian.' taking command comes down to knowledge, possession to the perfection of power, and protection to action. The one who unites these meanings is named Guardian. But only God - great and glorious - joins them absolutely and perfectly."

Reflection:
How do we benefit from knowing that Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ is Al-Muhaymin? Just as Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ is the overseer of all His creation, we need to be watchful over ourselves. Many people do not know themselves - they think they are their thoughts. We can let horrible or negative thoughts come to mind, but we don't have to identify with them. We can let it come and pass, just like how a river flows. Identification, thinking you are your thoughts, is like a beaver building a dam in the river. This only ends up damaging the surrounding environment.

Similarly, identification can be destructive, leading to either a depressive state or causing one to become arrogant or narcissistic. Instead, it is better to try and be indifferent. Take time to observe and watch your thoughts come and go. Ask where and why they may have arisen. Analyze their depths, and you may uncover secrets about yourself hiding in your psyche. Imam Ghazali recognizes this watching as a crucial step in reforming oneself, allowing one to become a guardian over his or her heart. Only selectively choosing thoughts that are constructive and letting everything else go.

Another way in which one can try to characterize themselves by this name is to be observant of those around you. People in passing will reveal certain pains and troubles. If you are watchful, you can exhibit a Prophetic tradition of the Prophet Muhammad ﷺ which was to care for the needs of those around him. In this sense, you are a guardian, being watchful and showing care and compassion to those around you.