The Honourer, The Bestower of Might.
He gives esteem to whoever He willed, hence there is no one to degrade Him; And He degrades whoever He willed, hence there is no one to give Him esteem. Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ is both al-Aziz (possessor of glory) and al-Mu’izz (giver of glory). He gives strength and honor to his obedient and loving servants. He gives material power to whoever he wills. He strengthens, glorifies, and makes it possible for someone or something to be respected.
From the root ‘-z-z which has the following classical Arabic connotations: to be mighty, potent, strong, powerful, high, elevated, noble, exalted, honorable, to be noble, glorious, illustrious, indomitable, invincible, respected, cherished, dear to, scarce, rare, precious, and/or unattainable.
The next two names in the series (i.e., al-Mu'izz and al-Mudhill) are closely related to the previous names. The emphasis of these attributes is on giving dominion or sovereignty or to take it away. As mentioned by Imam al-Ghazali, “true dominion consists in deliverance from the shame of need, the dominance of passion, and the disgrace of ignorance.”
We've also covered how varying scholars have different criteria for what consitutes as a name of Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ. al-Mu'izz is not fully recognized as an official name by certain scholars such as Ibn Uthaymeen, Ibn Hazm, and Ibn al-Wazir among others. As a rule of thumb, if the name contains prefix alif lam before the name then that's a clear indication of belonging as a name of Allah.
There are many quranic verses which describe Allah as being the one who one who gives izzah meaning strength, power, or dignity. You will see in the following verses where this attribute of being the bestower of might comes from. He gives dominion to whoever he pleases, but it doesn't explicitly use the name al-Mu'izz.
"Say, “O Allah, Owner of Sovereignty, You give sovereignty to whom You will and You take sovereignty away from whom You will. You honor whom You will and You humble whom You will. In Your hand is [all] good. Indeed, You are over all things competent." (Qur'an 3:26)
"So he began [the search] with their bags before the bag of his brother; then he extracted it from the bag of his brother. Thus did We plan for Joseph. He could not have taken his brother within the religion of the king except that Allah willed. We raise in degrees whom We will, but over every possessor of knowledge is one [more] knowing." (Qur'an 12:76)
To be among the honored Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ tells in Surah Hujurat verse 76 "the most noble of you in the sight of Allah is the most righteous of you. Indeed, Allah is Knowing and Acquainted."
"Whoever desires honor [through power] – then to Allah belongs all honor. To Him ascends good speech, and righteous work raises it. But they who plot evil deeds will have a severe punishment, and the plotting of those – it will perish." (35:10)
But seek, through that which Allah has given you, the home of the Hereafter; and [yet], do not forget your share of the world. And do good as Allah has done good to you. And desire not corruption in the land. Indeed, Allah does not like corrupters. (Quran 28:77)